773 Sentences With "bipartite" | Random Sentence Generator

Chandra's office offers a quick introduction to his bipartite mind.

From a constitutional perspective, it looked like a mosh pit of self-loathing members, politicians eager to be declared a functional non-entity in our tripartite (now bipartite) system.

Every chordal bipartite graph is a modular graph. The chordal bipartite graphs include the complete bipartite graphs and the bipartite distance-hereditary graphs.Chordal bipartite graphs, Information System on Graph Classes and their Inclusions, retrieved 2016-09-30.

The bipartite double cover of any graph G is a bipartite graph; both parts of the bipartite graph have one vertex for each vertex of G. A bipartite double cover is connected if and only if G is connected and non-bipartite., Theorem 3.4. The bipartite double cover is a special case of a double cover (a 2-fold covering graph). A double cover in graph theory can be viewed as a special case of a topological double cover.

A form of Wagner's theorem applies for bipartite minors: A bipartite graph G is a planar graph if and only if it does not have the utility graph K3,3 as a bipartite minor..

In , two other characterizations are mentioned: B is chordal bipartite if and only if every minimal edge separator induces a complete bipartite subgraph in B if and only if every induced subgraph is perfect elimination bipartite. Martin Farber has shown: A graph is strongly chordal if and only if the bipartite incidence graph of its clique hypergraph is chordal bipartite. ; , Theorem 3.4.1, p. 43.

In a nontrivial bipartite graph, the optimal number of colors is (by definition) two, and (since bipartite graphs are triangle-free) the maximum clique size is also two. Also, any induced subgraph of a bipartite graph remains bipartite. Therefore, bipartite graphs are perfect. In n-vertex bipartite graphs, a minimum clique cover takes the form of a maximum matching together with an additional clique for every unmatched vertex, with size n − M, where M is the cardinality of the matching.

A bipartite graph B = (X,Y,E) is chordal bipartite if and only if every induced subgraph of B has a maximum X-neighborhood ordering and a maximum Y-neighborhood ordering. Various results describe the relationship between chordal bipartite graphs and totally balanced neighborhood hypergraphs of bipartite graphs. , Theorems 8.2.5, 8.2.

Eulerian matroids were defined by as a generalization of the bipartite graphs, graphs in which every cycle has even size. A graphic matroid is bipartite if and only if it comes from a bipartite graph..

The bipartite Kneser graph can be formed as a bipartite double cover of in which one makes two copies of each vertex and replaces each edge by a pair of edges connecting corresponding pairs of vertices . The bipartite Kneser graph is the Desargues graph and the bipartite Kneser graph is a crown graph.

An alternative characterization of the bipartite graphs that may be formed by the bipartite double cover construction was obtained by .

Besides balance, there are alternative generalizations of bipartite graphs. A hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X (see bipartite hypergraph). Obviously every bipartite graph is 2-colorable. The properties of bipartiteness and balance do not imply each other.

Various problems such as Hamiltonian cycle, Steiner tree and Efficient Domination remain NP-complete on chordal bipartite graphs. Various other problems which can be solved efficiently for bipartite graphs can be solved more efficiently for chordal bipartite graphs as discussed in .

In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph.

In the above example, each vertex of H has exactly 2 preimages in C. Hence H is a 2-fold cover or a double cover of C. For any graph G, it is possible to construct the bipartite double cover of G, which is a bipartite graph and a double cover of G. The bipartite double cover of G is the tensor product of graphs G × K2: :File:Covering-graph-2.svg If G is already bipartite, its bipartite double cover consists of two disjoint copies of G. A graph may have many different double covers other than the bipartite double cover.

131–145 In case of regular bipartite graphs equality holds. Subcubic bipartite graphs admit an interval incidence coloring using four, five or six colors. They have also proved incidence 5-colorability can be decided in linear time for bipartite graphs with ∆(G) = 4.

A similar characterization holds for the closed neighborhood hypergraph: A graph is strongly chordal if and only if the bipartite incidence graph of its closed neighborhood hypergraph is chordal bipartite. Another result found by Elias Dahlhaus is: A bipartite graph B = (X,Y,E) is chordal bipartite if and only if the split graph resulting from making X a clique is strongly chordal., Corollary 8.3.2, p. 129.

The hitchcock problem asks for such a subgraph minimizing the sum of the costs on each edge which are given for the complete bipartite graph. A further generalization is the f-factor problem for bipartite graphs, i.e. for a given bipartite graph one searches for a subgraph possessing a certain degree sequence. The problem uniform sampling a bipartite graph to a fixed degree sequence is to construct a solution for the bipartite realization problem with the additional constraint that each such solution comes with the same probability.

For planar graphs, the properties of being Eulerian and bipartite are dual: a planar graph is Eulerian if and only if its dual graph is bipartite. As Welsh showed, this duality extends to binary matroids: a binary matroid is Eulerian if and only if its dual matroid is a bipartite matroid, a matroid in which every circuit has even cardinality.. For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down. For instance, the uniform matroid U{}^2_6 is Eulerian but its dual U{}^4_6 is not bipartite, as its circuits have size five. The self-dual uniform matroid U{}^3_6 is bipartite but not Eulerian.

6, pp. 126–127. A characterization of chordal bipartite graphs in terms of intersection graphs related to hypergraphs is given in. A bipartite graph is chordal bipartite if and only if its adjacency matrix is totally balanced if and only if the adjacency matrix is Gamma-free.

Aharoni and Berger generalized Drisko's theorem to any bipartite graph, namely: any family of 2n-1 matchings of size n in a bipartite graph has a rainbow matching of size n. Aharoni, Kotlar and Ziv showed that Drisko's extremal example is unique in any bipartite graph.

Many examples of problems with checkable algorithms come from graph theory. For instance, a classical algorithm for testing whether a graph is bipartite would simply output a Boolean value: true if the graph is bipartite, false otherwise. In contrast, a certifying algorithm might output a 2-coloring of the graph in the case that it is bipartite, or a cycle of odd length if it is not. Any graph is bipartite if and only if it can be 2-colored, and non- bipartite if and only if it contains an odd cycle.

Therefore, every modular graph is a bipartite graph. The modular graphs contain as a special case the median graphs, in which every triple of vertices has a unique median; median graphs are related to distributive lattices in the same way that modular graphs are related to modular lattices. However, the modular graphs also include other graphs such as the complete bipartite graphs where the medians are not unique: when the three vertices , , and all belong to one side of the bipartition of a complete bipartite graph, every vertex on the other side is a median. Every chordal bipartite graph (a class of graphs that includes the complete bipartite graphs and the bipartite distance-hereditary graphs) is modular.

This statement can be also applied to obtain an upper bound of the Ramsey number of a degenerate bipartite graphs. If r is a fixed integer, then for every bipartite r-degenerate bipartite graph G on n vertices, the Ramsey number r(G) is of the order n^{1+o(1)}.

1, p. 43. A better name would be weakly chordal and bipartite since chordal bipartite graphs are in general not chordal as the induced cycle of length 4 shows.

The Erdős–Szekeres theorem is an easy consequence of this statement. Kőnig's theorem in graph theory states that a minimum vertex cover in a bipartite graph corresponds to a maximum matching, and vice versa; it can be interpreted as the perfection of the complements of bipartite graphs. Another theorem about bipartite graphs, that their chromatic index equals their maximum degree, is equivalent to the perfection of the line graphs of bipartite graphs.

A chordal domino is a chordal graph in which every vertex belongs to at most two maximal cliques. the complements of comparability graphs, and bipartite distance-hereditary graphs.. It follows immediately that it is also NP- complete for the graph families that contain the bipartite distance-hereditary graphs, including the bipartite graphs, chordal bipartite graphs, distance- hereditary graphs, and circle graphs. However, the pathwidth may be computed in linear time for trees and forests,.; ; ; ; ; ; .

In graph theory, the bipartite double cover of an undirected graph G is a bipartite covering graph of G, with twice as many vertices as G. It can be constructed as the tensor product of graphs, G × K2. It is also called the Kronecker double cover, canonical double cover or simply the bipartite double of G. It should not be confused with a cycle double cover of a graph, a family of cycles that includes each edge twice. In a connected graph that is not bipartite, only one double cover is bipartite, but when the graph is bipartite or disconnected there may be more than one. For this reason, Tomaž Pisanski has argued that the name "bipartite double cover" should be deprecated in favor of the "canonical double cover" or "Kronecker cover" names, which are unambiguous.

The halved cube graph of order 4, obtained as the bipartite half of an order-4 hypercube graph In graph theory, the bipartite half or half-square of a bipartite graph G = (U,V,E) is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, U) and in which there is an edge uiuj for each two vertices ui and uj in U that are at distance two from each other in G.. That is, in a more compact notation, the bipartite half is G2[U] where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph. For instance, the bipartite half of the complete bipartite graph Kn,n is the complete graph Kn and the bipartite half of the hypercube graph is the halved cube graph. When G is a distance-regular graph, its two bipartite halves are both distance-regular.. For instance, the halved Foster graph is one of finitely many degree-6 distance-regular locally linear graphs. The map graphs, that is, the intersection graphs of interior- disjoint simply-connected regions in the plane, are exactly the bipartite halves of bipartite planar graphs..

Bipartite maximum matchings can be approximated arbitrarily accurately in constant time by distributed algorithms; in contrast, approximating the minimum vertex cover of a bipartite graph requires at least logarithmic time..

This idea was based on previous conjectured structural decompositions of similar type that would have implied the strong perfect graph conjecture but turned out to be false.; ; ; , section 4.6 "The first conjectures". The five basic classes of perfect graphs that form the base case of this structural decomposition are the bipartite graphs, line graphs of bipartite graphs, complementary graphs of bipartite graphs, complements of line graphs of bipartite graphs, and double split graphs. It is easy to see that bipartite graphs are perfect: in any nontrivial induced subgraph, the clique number and chromatic number are both two and therefore both equal.

A distance-hereditary graph is bipartite if and only if it is triangle-free. Bipartite distance- hereditary graphs can be built up from a single vertex by adding only pendant vertices and false twins, since any true twin would form a triangle, but the pendant vertex and false twin operations preserve bipartiteness. Every bipartite distance-hereditary graph is chordal bipartite and modular.Bipartite distance-hereditary graphs, Information System on Graph Classes and their Inclusions, retrieved 2016-09-30.

The bipartite realization problem is a classical decision problem in graph theory, a branch of combinatorics. Given two finite sequences (a_1,\dots,a_n) and (b_1,\dots,b_n) of natural numbers, the problem asks whether there is labeled simple bipartite graph such that (a_1,\dots,a_n),(b_1,\dots,b_n) is the degree sequence of this bipartite graph.

Similar problems describe the degree sequences of simple graphs and simple directed graphs. The first problem is the so-called graph realization problem. The second is known as the digraph realization problem. The bipartite realization problem is equivalent to the question, if there exists a labeled bipartite subgraph of a complete bipartite graph to a given degree sequence.

Another important tool are clique separators as described by Tarjan. Kőnig's theorem implies that in a bipartite graph the maximum independent set can be found in polynomial time using a bipartite matching algorithm.

In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ∪ V, E) that is convex over both U and V is said to be biconvex or doubly convex.

It is automatically true when the graph contains an odd cycle, because the independent set of all heavy vertices cannot cover all the edges of the cycle. Therefore, the more interesting case of the conjecture is for bipartite graphs, which have no odd cycles. Another equivalent formulation of the conjecture is that, in every bipartite graph, there exist two vertices, one on each side of the bipartition, such that each of these two vertices belongs to at most half of the graph's maximal independent sets. This conjecture is known to hold for chordal bipartite graphs, bipartite series-parallel graphs, and bipartite graphs of maximum degree three.

However, the icosahedron is not bipartite, so it is not the bipartite double cover of K6. Instead, it can be obtained as the orientable double cover of an embedding of K6 on the projective plane.

Macarena Cabrillana earned her spot via a Bipartite Commission Invitation place .

Bipartite network projection is an extensively used method for compressing information about bipartite networks."Bipartite network projection and personal recommendation" by Tao Zhou, Jie Ren, Matúš Medo and Yi-Cheng Zhang in PHYSICAL REVIEW E 76(4): 046115 (2007) Since the one-mode projection is always less informative than the original bipartite graph, an appropriate method for weighting network connections is often required. Optimal weighting methods reflect the nature of the specific network, conform to the designer's objectives and aim at minimizing information loss.

Here, we use properties of graph entropy to provide a simple proof that a complete graph G on n vertices cannot be expressed as the union of fewer than \log_2 n bipartite graphs. Proof By monotonicity, no bipartite graph can have graph entropy greater than that of a complete bipartite graph, which is bounded by 1. Thus, by sub-additivity, the union of k bipartite graphs cannot have entropy greater than k. Now let G = (V, E) be a complete graph on n vertices.

See Zarankiewicz problem for more on the extremal functions of bipartite graphs.

An Eulerian graph is one in which all vertices have even degree; Eulerian graphs may be disconnected. For planar graphs, the properties of being bipartite and Eulerian are dual: a planar graph is bipartite if and only if its dual graph is Eulerian. As Welsh showed, this duality extends to binary matroids: a binary matroid is bipartite if and only if its dual matroid is an Eulerian matroid, a matroid that can be partitioned into disjoint circuits. For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down.

In graph theory, a balanced hypergraph is a hypergraph that has several properties analogous to that of a bipartite graph. Balanced hypergraphs were introduced by Berge as a natural generalization of bipartite graphs. He provided two equivalent definitions.

For instance, the uniform matroid U{}^4_6 is non-bipartite but its dual U{}^2_6 is Eulerian, as it can be partitioned into two 3-cycles. The self-dual uniform matroid U{}^3_6 is bipartite but not Eulerian.

Kőnig's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover, maximum independent set, and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.

Finally, the contraction criterion generalizes immediately from the bipartite to the multipartite case.

These types of NLSs can be further classified as either monopartite or bipartite. The major structural differences between the two is that the two basic amino acid clusters in bipartite NLSs are separated by a relatively short spacer sequence (hence bipartite - 2 parts), while monopartite NLSs are not. The first NLS to be discovered was the sequence PKKKRKV in the SV40 Large T-antigen (a monopartite NLS). The NLS of nucleoplasmin, KR[PAATKKAGQA]KKKK, is the prototype of the ubiquitous bipartite signal: two clusters of basic amino acids, separated by a spacer of about 10 amino acids.

The line graph of a bipartite graph is perfect (see Kőnig's theorem), but need not be bipartite as the example of the claw graph shows. The line graphs of bipartite graphs form one of the key building blocks of perfect graphs, used in the proof of the strong perfect graph theorem.. See also . A special case of these graphs are the rook's graphs, line graphs of complete bipartite graphs. Like the line graphs of complete graphs, they can be characterized with one exception by their numbers of vertices, numbers of edges, and number of shared neighbors for adjacent and non-adjacent points.

In mathematics, a bipartite matroid is a matroid all of whose circuits have even size.

The number of perfect matchings of a bipartite graph can be calculated using the principle.

In the mathematical fields of graph theory and combinatorial optimization, the bipartite dimension or biclique cover number of a graph G = (V, E) is the minimum number of bicliques (that is complete bipartite subgraphs), needed to cover all edges in E. A collection of bicliques covering all edges in G is called a biclique edge cover, or sometimes biclique cover. The bipartite dimension of G is often denoted by the symbol d(G).

A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and (x1, e1) are connected with an edge if and only if vertex x1 is contained in edge e1 in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. This bipartite graph is also called incidence graph.

It was demonstrated by that it is NP-complete to determine whether \chi_s(G) \leq 3, even when G is a graph that is both planar and bipartite. showed that finding an optimal star coloring is NP-hard even when G is a bipartite graph.

If there were fewer than n-1 complete bipartite graphs, the system of equations would have fewer than n equations in n unknowns and would have a nontrivial solution, a contradiction. So the number of complete bipartite graphs must be at least n-1.

The elastic, delicate radular membrane may be a single tongue, or may split into two (bipartite).

Chordal bipartite graphs have various characterizations in terms of perfect elimination orderings, hypergraphs and matrices. They are closely related to strongly chordal graphs. By definition, chordal bipartite graphs have a forbidden subgraph characterization as the graphs that do not contain any induced cycle of length 3 or of length at least 5 (so-called holes) as an induced subgraph. Thus, a graph G is chordal bipartite if and only if G is triangle-free and hole-free.

Chordal bipartite graphs can be recognized in time for a graph with n vertices and m edges.; ; ; .

A bipartite graph is (a, b)-biregular if everyvertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. Hansen proved that every bipartite graph G with ∆(G) ≤ 3 is interval colorable.

A modillioned cornice conceals its roof. Windows are square-headed bipartite timber sashes with stone sills and lintels.

In graph theory a process graph or P-graph is a directed bipartite graph used in workflow modeling.

Mivar network representation in the form of a bipartite directed graph Mivar network is a method for representing objects of the subject domain and their processing rules in the form of a bipartite directed graph consisting of objects and rules. A Mivar network is a bipartite graph that can be described in the form of a two-dimensional matrix, in that records information about the subject domain of the current task. Generally, mivar networks provide formalization and representation of human knowledge in the form of a connected multidimensional space. That is, a mivar network is a method of representing a piece of mivar space information in the form of a bipartite, directed graph.

This entanglement measure is a generalization of the entanglement of assistance and was constructed in the context of spin chains. Namely, one chooses two spins and performs LOCC operations that aim at obtaining the largest possible bipartite entanglement between them (measured according to a chosen entanglement measure for two bipartite states).

Garey and Johnson have shown that computing a maximum matching is NP- complete even for edge-colored bipartite graphs.

Nicholas Taylor and David Wagner qualified via the standard route. Bryan Barten qualified via a Bipartite Commission Invitation place.

Some classes of matroid have been defined from well-known families of graphs, by phrasing a characterization of these graphs in terms that make sense more generally for matroids. These include the bipartite matroids, in which every circuit is even, and the Eulerian matroids, which can be partitioned into disjoint circuits. A graphic matroid is bipartite if and only if it comes from a bipartite graph and a graphic matroid is Eulerian if and only if it comes from an Eulerian graph. Within the graphic matroids (and more generally within the binary matroids) these two classes are dual: a graphic matroid is bipartite if and only if its dual matroid is Eulerian, and a graphic matroid is Eulerian if and only if its dual matroid is bipartite.. Graphic matroids are one- dimensional rigidity matroids, matroids describing the degrees of freedom of structures of rigid beams that can rotate freely at the vertices where they meet.

This was one of the results that motivated the initial definition of perfect graphs.. Perfection of the complements of line graphs of perfect graphs is yet another restatement of Kőnig's theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of Kőnig, that every bipartite graph has an edge coloring using a number of colors equal to its maximum degree. According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem..

Up to constant factors, z(n; t) also bounds the number of edges in an n-vertex graph (not required to be bipartite) that has no Kt,t subgraph. For, in one direction, a bipartite graph with z(n; t) edges and with n vertices on each side of its bipartition can be reduced to a graph with n vertices and (in expectation) z(n; t)/4 edges, by choosing n/2 vertices uniformly at random from each side. In the other direction, a graph with n vertices and no Kt,t can be transformed into a bipartite graph with n vertices on each side of its bipartition, twice as many edges, and still no Kt,t by taking its bipartite double cover., Theorem 2.3, p. 310.

A signal is said to be bipartite if the sequence of intervals start with a singular interval - i.e. a closed interval whose lower and upper bound are equal, hence a set which is a singleton. And if the sequence alternate between singular intervals and open intervals. Each signal is equivalent to a bipartite signal.

Map graphs can be represented combinatorially as the "half-squares of planar bipartite graphs". That is, let be a planar bipartite graph, with bipartition . The square of is another graph on the same vertex set, in which two vertices are adjacent in the square when they are at most two steps apart in . The half- square or bipartite half is the induced subgraph of one side of the bipartition (say ) in the square graph: its vertex set is and it has an edge between each two vertices in that are two steps apart in .

The number of matchings in a graph is known as the Hosoya index of the graph. It is #P-complete to compute this quantity, even for bipartite graphs.Leslie Valiant, The Complexity of Enumeration and Reliability Problems, SIAM J. Comput., 8(3), 410–421 It is also #P-complete to count perfect matchings, even in bipartite graphs, because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix.

As a simple example, suppose that a set P of people are all seeking jobs from among a set of J jobs, with not all people suitable for all jobs. This situation can be modeled as a bipartite graph (P,J,E) where an edge connects each job-seeker with each suitable job., Application 12.1 Bipartite Personnel Assignment, pp. 463–464. A perfect matching describes a way of simultaneously satisfying all job-seekers and filling all jobs; Hall's marriage theorem provides a characterization of the bipartite graphs which allow perfect matchings.

Every complete bipartite graph K_{a,b} is (b,a)-biregular. The rhombic dodecahedron is another example; it is (3,4)-biregular..

A 14-vertex half graph In graph theory, a branch of mathematics, a half graph is a special type of bipartite graph. These graphs are called the half graphs because they have approximately half of the edges of a complete bipartite graph on the same vertices. The name was given to these graphs by Paul Erdős and András Hajnal.

In the mathematical area of graph theory, a chordal bipartite graph is a bipartite graph B = (X,Y,E) in which every cycle of length at least 6 in B has a chord, i.e., an edge that connects two vertices that are a distance > 1 apart from each other in the cycle. , p. 261, , Definition 3.4.

This protein is predicted to be a nuclear protein. There appears to be a bipartite nuclear localization sequence beginning at position 80.

Bipartite networks are a particular class of complex networks, whose nodes are divided into two sets X and Y, and only connections between two nodes in different sets are allowed. For the convenience of directly showing the relation structure among a particular set of nodes, bipartite networks are usually compressed by one-mode projection. This means that the ensuing network contains nodes of only either of the two sets, and two X (or, alternatively, Y) nodes are connected only if when they have at least one common neighboring Y (or, alternatively, X) node. "Possible projections of a simple bipartite network" The simplest method involves projecting the bipartite network onto an unweighted network, without taking into account the topology of the network or the frequency of sharing a connection to the elements of the opposing set.

For instance, the complete bipartite graph K3,6 is 1-planar because it is a subgraph of K1,1,1,6, but K3,7 is not 1-planar..

4, 367–377.M.I. Isaev, Asymptotic number of Eulerian circuits in complete bipartite graphs (in Russian), Proc. 52-nd MFTI Conference (2009), Moscow.

As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.

More precisely, the problem is polynomial-time if the graph is 2-colorable, that is, it is bipartite, and is NP-complete otherwise.

Reducing Minimum weight bipartite matching to minimum cost max flow problem Given a bipartite graph G = (A ∪ B, E), the goal is to find the maximum cardinality matching in G that has minimum cost. Let w: E → R be a weight function on the edges of E. The minimum weight bipartite matching problem or assignment problem is to find a perfect matching M ⊆ E whose total weight is minimized. The idea is to reduce this problem to a network flow problem. Let G′ = (V′ = A ∪ B, E′ = E). Assign the capacity of all the edges in E′ to 1.

In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set... Electronic edition, page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher.. Llull himself had made similar drawings of complete graphs three centuries earlier..

Hypohamiltonian snarks do not have a partition into matchings of this type, but conjectures that the edges of any hypohamiltonian snark may be used to form six matchings such that each edge belongs to exactly two of the matchings. This is a special case of the Berge–Fulkerson conjecture that any snark has six matchings with this property. Hypohamiltonian graphs cannot be bipartite: in a bipartite graph, a vertex can only be deleted to form a Hamiltonian subgraph if it belongs to the larger of the graph's two color classes. However, every bipartite graph occurs as an induced subgraph of some hypohamiltonian graph..

When n ≡ 0 (mod 4), it is not bipartite. The endpoints of each rung are an even distance apart in the initial cycle, so adding each rung creates an odd cycle. In this case, because the graph is 3-regular but not bipartite, by Brooks' theorem it has chromatic number 3. show that the Möbius ladders are uniquely determined by their Tutte polynomials.

Similar problems describe the degree sequences of simple graphs, simple directed graphs with loops, and simple bipartite graphs. The first problem is the so-called graph realization problem. The second and third one are equivalent and are known as the bipartite realization problem. gaves a characterization for directed multigraphs with a bounded number of parallel arcs and loops to a given degree sequence.

This is because most multipartite viruses infect plants or fungi, which are eukaryotes, and most eukaryotic viruses are RNA viruses.Fermin 2018, pp. 35–46 The family Pleolipoviridae varies as some viruses are monopartite ssDNA while others are bipartite with one segment being ssDNA and the other dsDNA. Viruses in the ssDNA plant virus family Geminiviridae likewise vary between being monopartite and bipartite.

Every complete graph is a comparability graph, the comparability graph of a total order. All acyclic orientations of a complete graph are transitive. Every bipartite graph is also a comparability graph. Orienting the edges of a bipartite graph from one side of the bipartition to the other results in a transitive orientation, corresponding to a partial order of height two.

The genome consists of 2.8 kb of circular single stranded DNA (ssDNA). Unlike the majority of virus genomes in the Begomovirus family which are bipartite, the SPLCV is monopartite. It contains only a single genomic component, which is similar to DNA-A of bipartite viruses. DNA-A typically encodes products for DNA replication, controls gene expression, and controls insect transmission.

A spanning Sachs subgraph, also called a {1,2}-factor, is a Sachs subgraph in which every vertex of the given graph is incident to an edge of the subgraph. The union of two perfect matchings is always a bipartite spanning Sachs subgraph, but in general Sachs subgraphs are not restricted to being bipartite. Some authors use the term "Sachs subgraph" to mean only spanning Sachs subgraphs.

Ellingham, M. N. and Horton, J. D. "Non-Hamiltonian 3-Connected Cubic Bipartite Graphs." J. Combin. Th. Ser. B 34, 350-353, 1983. In 1989, Georges' graph, the smallest currently- known non-Hamiltonian 3-connected cubic bipartite graph was discovered, containing 50 vertices.. As a non-Hamiltonian cubic graph with many long cycles, the Horton graph provides good benchmark for programs that search for Hamiltonian cycles.

An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by , describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs.

Partition of the edges of the complete graph K_6 into five complete bipartite subgraphs: K_{2,2} (light red), K_{2,3} (light blue), K_{1,3} (yellow), and two copies of K_{1,1} (dark red and dark blue). According to the Graham–Pollak theorem, a partition into fewer than five complete bipartite subgraphs is not possible. In graph theory, the Graham–Pollak theorem states that the edges of an n-vertex complete graph cannot be partitioned into fewer than n-1 complete bipartite graphs. It was first published by Ronald Graham and Henry O. Pollak in two papers in 1971 and 1972, in connection with an application to telephone switching circuitry.

A matching in a graph is a subset of its edges, no two of which share an endpoint. Polynomial time algorithms are known for many algorithmic problems on matchings, including maximum matching (finding a matching that uses as many edges as possible), maximum weight matching, and stable marriage.. In many cases, matching problems are simpler to solve on bipartite graphs than on non- bipartite graphs,, p. 463: "Nonbipartite matching problems are more difficult to solve because they do not reduce to standard network flow problems." and many matching algorithms such as the Hopcroft–Karp algorithm for maximum cardinality matching. work correctly only on bipartite inputs.

A star Sk is the complete bipartite graph K1,k. The star S3 is called the claw graph. The star graphs S3, S4, S5 and S6.

A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. Odd cycle transversal is an NP-complete algorithmic problem that asks, given a graph G = (V,E) and a number k, whether there exists a set of k vertices whose removal from G would cause the resulting graph to be bipartite. The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k.. The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles. Hence, to delete vertices from a graph in order to obtain a bipartite graph, one needs to "hit all odd cycle", or find a so-called odd cycle transversal set.

A uniform matroid U{}^r_n is bipartite if and only if r is an odd number, because the circuits in such a matroid have size r+1.

The sequences are matched up by using a heuristic algorithm for maximizing the score globally, rather than locally, in a bipartite matching (see complete bipartite graph). And then it calculates the statistical significance of each match. Cutoffs are made per position and Ks values are set to prevent false "orthologs" from being grouped together. “Paralogs” are identified by finding sequences that are closer within species than other species.

In graph-theoretic mathematics, a biregular graph. or semiregular bipartite graph. is a bipartite graph G=(U,V,E) for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be (x,y)-biregular.

When G is bipartite, there are no odd cycles in G. In that case, blossoms will never be found and one can simply remove lines B20 - B24 of the algorithm. The algorithm thus reduces to the standard algorithm to construct maximum cardinality matchings in bipartite graphs where we repeatedly search for an augmenting path by a simple graph traversal: this is for instance the case of the Ford–Fulkerson algorithm.

A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. In graph theory, an odd cycle transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd cycle transversal from a graph leaves a bipartite graph as the remaining induced subgraph.

All such graphs are bipartite, and hence can be colored with only two colors. An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example. Every edge- transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.. The vertex connectivity of an edge-transitive graph always equals its minimum degree.

In a weighted bipartite graph, the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching, or the assignment problem. The Hungarian algorithm solves the assignment problem and it was one of the beginnings of combinatorial optimization algorithms. It uses a modified shortest path search in the augmenting path algorithm.

Since any 0–1 matrix is the biadjacency matrix of some bipartite graph, Valiant's theorem impliesDexter Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991.

The bipartite double cover of G has two vertices ui and wi for each vertex vi of G. Two vertices ui and wj are connected by an edge in the double cover if and only if vi and vj are connected by an edge in G. For instance, below is an illustration of a bipartite double cover of a non-bipartite graph G. In the illustration, each vertex in the tensor product is shown using a color from the first term of the product (G) and a shape from the second term of the product (K2); therefore, the vertices ui in the double cover are shown as circles while the vertices wi are shown as squares. :Image:Covering-graph-2.svg The bipartite double cover may also be constructed using adjacency matrices (as described below) or as the derived graph of a voltage graph in which each edge of G is labeled by the nonzero element of the two-element group.

See for proofs showed that Barnette's conjecture is equivalent to a superficially stronger statement, that for every two edges e and f on the same face of a bipartite cubic polyhedron, there exists a Hamiltonian cycle that contains e but does not contain f. Clearly, if this statement is true, then every bipartite cubic polyhedron contains a Hamiltonian cycle: just choose e and f arbitrarily. In the other directions, Kelmans showed that a counterexample could be transformed into a counterexample to the original Barnette conjecture. Barnette's conjecture is also equivalent to the statement that the vertices of the dual of every cubic bipartite polyhedral graph can be partitioned into two subsets whose induced subgraphs are trees.

The first bona fide radula dates to the Early Cambrian, although trace fossils from the earlier Ediacaran have been suggested to have been made by the radula of the organism Kimberella. A so-called radula from the early Cambrian was discovered in 1974, this one preserved with fragments of the mineral ilmenite suspended in a quartz matrix, and showing similarities to the radula of the modern cephalopod Sepia. However, this was since re-interpreted as Salterella. [/Volborthella?] Based on the bipartite nature of the radular dentition pattern in solenogasters, larval gastropods and larval polyplacophora, it has been postulated that the ancestral mollusc bore a bipartite radula (although the radular membrane may not have been bipartite).

For some graphs, such as bipartite graphs and high- degree planar graphs, the number of colors is always , and for multigraphs, the number of colors may be as large as . There are polynomial time algorithms that construct optimal colorings of bipartite graphs, and colorings of non- bipartite simple graphs that use at most colors; however, the general problem of finding an optimal edge coloring is NP-hard and the fastest known algorithms for it take exponential time. Many variations of the edge coloring problem, in which an assignments of colors to edges must satisfy other conditions than non-adjacency, have been studied. Edge colorings have applications in scheduling problems and in frequency assignment for fiber optic networks.

The open-shop scheduling problem can be solved in polynomial time for instances that have only two workstations or only two jobs. It may also be solved in polynomial time when all nonzero processing times are equal: in this case the problem becomes equivalent to edge coloring a bipartite graph that has the jobs and workstations as its vertices, and that has an edge for every job-workstation pair that has a nonzero processing time. The color of an edge in the coloring corresponds to the segment of time at which a job-workstation pair is scheduled to be processed. Because the line graphs of bipartite graphs are perfect graphs, bipartite graphs may be edge-colored in polynomial time.

For example, there are exactly 2 nodes in C that are mapped to the blue node in H. However, C is not a bipartite double cover of H or any other graph; it is not a bipartite graph. If we replace one triangle by a square in H the resulting graph has four distinct double covers. Two of them are bipartite but only one of them is the Kronecker cover. :Image:Covering-graph-4.svg As another example, the graph of the icosahedron is a double cover of the complete graph K6; to obtain a covering map from the icosahedron to K6, map each pair of opposite vertices of the icosahedron to a single vertex of K6.

A large chalazion ca. 20 minutes upon excision. This bipartite chalazion was removed via two separate incisions. Further along the lower eyelid, signs of chronic inflammation (Blepharitis) are visible.

There are other natural generalizations of bipartite graphs. A hypergraph is called balanced if it is essentially 2-colorable, and remains essentially 2-colorable upon deleting any number of vertices (see Balanced hypergraph). The properties of bipartiteness and balance do not imply each other. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: > { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}.

The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph. The clique cover problem concerns finding as few cliques as possible that include every vertex in the graph. A related concept is a biclique, a complete bipartite subgraph. The bipartite dimension of a graph is the minimum number of bicliques needed to cover all the edges of the graph.

If edges are required to be drawn as straight line segments, rather than arbitrary curves, then some graphs need more crossings than they would when drawn with curved edges. However, the upper bound established by Zarankiewicz for the crossing numbers of complete bipartite graphs can be achieved using only straight edges. Therefore, if the Zarankiewicz conjecture is correct, then the complete bipartite graphs have rectilinear crossing numbers equal to their crossing numbers.

A fundamental problem in combinatorial optimization is finding a maximum matching. This problem has various algorithms for different classes of graphs. In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.

There is a maximum of 1100 athletics qualification slots available; 660 male and 440 female. In general, qualification slots are awarded to the athlete's National Paralympic Committee (NPC), not the individual athlete. The exception is with the Bipartite Commission Invitation slots, which are awarded to the individual athlete, not the NPC. Each NPC can allocated a maximum of 48 male qualification slots and 32 female allocation slots (excluding Bipartite Commission Invitation slots).

The arrangement of interactions within complex bipartite networks may be nested as well. More specifically, bipartite ecological and organisational networks of mutually beneficial interactions were found to have a nested structure. This structure promotes indirect facilitation and a system's capacity to persist under increasingly harsh circumstances as well as the potential for large-scale systemic regime shifts. ;Dynamic network of multiplicity :As well as coupling rules, the dynamic network of a complex system is important.

It is an isometric particle with a bipartite RNA genome. The virus has a wide host rangePrice, Am. J. Bot. 27: 530, 1940. that includes field grown crops, ornamentals and weeds.

The classes of graphs with bounded degeneracy and of nowhere dense graphs are both included in the biclique-free graphs, graph families that exclude some complete bipartite graph as a subgraph .

If M is a binary matroid, then so is its dual, and so is every minor of M. Additionally, the direct sum of binary matroids is binary. define a bipartite matroid to be a matroid in which every circuit has even cardinality, and an Eulerian matroid to be a matroid in which the elements can be partitioned into disjoint circuits. Within the class of graphic matroids, these two properties describe the matroids of bipartite graphs and Eulerian graphs (not-necessarily-connected graphs in which all vertices have even degree), respectively. For planar graphs (and therefore also for the graphic matroids of planar graphs) these two properties are dual: a planar graph or its matroid is bipartite if and only if its dual is Eulerian.

In graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u1, u2, ..., un} and {v1, v2, ..., vn} and with an edge from ui to vj whenever i ≠ j. The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product Kn × K2, as the complement of the Cartesian direct product of Kn and K2, or as a bipartite Kneser graph Hn,1 representing the 1-item and (n − 1)-item subsets of an n-item set, with an edge between two subsets whenever one is contained in the other.

In computer science, a Petri net is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.. In projective geometry, Levi graphs are a form of bipartite graph used to model the incidences between points and lines in a configuration.

In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every k divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by n, the length of that cycle.

The Gale–Ryser theorem is a result in graph theory and combinatorial matrix theory, two branches of combinatorics. It provides one of two known approaches to solving the bipartite realization problem, i.e. it gives a necessary and sufficient condition for two finite sequences of natural numbers to be the degree sequence of a labeled simple bipartite graph; a sequence obeying these conditions is called "bigraphic". It is an analog of the Erdős–Gallai theorem for simple graphs.

Studies have suggested that most of the expression is found in the cytoplasm of the cell, but there is also evidence of expression in the nucleus. Expression in the nucleus may be supported by the fact that the rat homolog of the SPATS1 gene was experimentally found to have a probable bipartite nuclear localization signal. In addition, bioinformatic tools have identified a bipartite nuclear localization signal with high probability in the human protein at amino acids 174 - 191.

Process network synthesis (PNS) is a method to represent a process structure in a 'directed bipartite graph'. Process network synthesis uses the P-graph method to create a process structure. The scientific aim of this method is to find optimum structures. Process network synthesis uses a bipartite graph method P-graphP-graph method and employs combinatorial rules to find all feasible network solutions (maximum structure) and links raw materials to desired products related to the given problem.

However, if there exists at least one odd cycle, then no 2-edge-coloring is possible. That is, a graph with is of class one if and only if it is bipartite.

Both signals are recognized by importin α. Importin α contains a bipartite NLS itself, which is specifically recognized by importin β. The latter can be considered the actual import mediator. Chelsky et al.

190 and 195. For bipartite graphs, the equivalence between vertex cover and maximum matching described by Kőnig's theorem allows the bipartite vertex cover problem to be solved in polynomial time. For tree graphs, an algorithm finds a minimal vertex cover in polynomial time by finding the first leaf in the tree and adding its parent to the minimal vertex cover, then deleting the leaf and parent and all associated edges and continuing repeatedly until no edges remain in the tree.

Bipartite Heawood graph. Points are represented by vertices of one color and lines by vertices of the other color. As with any incidence structure, the Levi graph of the Fano plane is a bipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are incident. This particular graph is a connected cubic graph (regular of degree 3), has girth 6 and each part contains 7 vertices.

The tree-depths of the complete graph K4 and the complete bipartite graph K3,3 are both four, while the tree-depth of the path graph P7 is three. The tree-depth of a complete graph equals its number of vertices. For, in this case, the only possible forest F for which every pair of vertices are in an ancestor-descendant relationship is a single path. Similarly, the tree-depth of a complete bipartite graph Kx,y is min(x,y) + 1\.

A partition of the complete bipartite graph K4,4 into three forests, showing that it has arboricity three. The figure shows the complete bipartite graph K4,4, with the colors indicating a partition of its edges into three forests. K4,4 cannot be partitioned into fewer forests, because any forest on its eight vertices has at most seven edges, while the overall graph has sixteen edges, more than double the number of edges in a single forest. Therefore, the arboricity of K4,4 is three.

If a graph does not have finite chromatic number, then the De Bruijn–Erdős theorem implies that it must contain finite subgraphs of every possible finite chromatic number. Researchers have also investigated other conditions on the subgraphs that are forced to occur in this case. For instance, unboundedly chromatic graphs must also contain every possible finite bipartite graph as a subgraph. However, they may have arbitrarily large odd girth, and therefore they may avoid any finite set of non-bipartite subgraphs.

As a distance-regular graph with odd girth k and diameter (k − 1)/2, the folded cubes of odd order are examples of generalized odd graphs.. When k is odd, the bipartite double cover of the order-k folded cube is the order-k cube from which it was formed. However, when k is even, the order-k cube is a double cover but not the bipartite double cover. In this case, the folded cube is itself already bipartite. Folded cube graphs inherit from their hypercube subgraphs the property of having a Hamiltonian cycle, and from the hypercubes that double cover them the property of being a distance- transitive graph.. When k is odd, the order-k folded cube contains as a subgraph a complete binary tree with 2k − 1 nodes.

The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices but has no complete bipartite subgraphs of a given size.. Reprint of 1978 Academic Press edition, . It belongs to the field of extremal graph theory, a branch of combinatorics, and is named after the Polish mathematician Kazimierz Zarankiewicz, who proposed several special cases of the problem in 1951.. As cited by . The Kővári–Sós–Turán theorem, named after Tamás Kővári, Vera T. Sós, and Pál Turán, provides an upper bound on the solution to the Zarankiewicz problem. When the forbidden complete bipartite subgraph has one side with at most three vertices, this bound has been proven to be within a constant factor of the correct answer.

Since bipartite networks with largely different structures can have exactly the same one-mode representation in this case, a lucid illustration of the original network topology usually requires the use of some weighting method.

Stochastic Block Model have been recognised to be a topic model on bipartite networks . In a network of documents and words, Stochastic Block Model can identify topics: group of words with a similar meaning.

Marston Conder has compiled a Complete list of all connected edge-transitive graphs on up to 47 vertices and a Complete list of all connected edge-transitive bipartite graphs on up to 63 vertices.

This in turn yields in-place algorithms for problems such as determining if a graph is bipartite or testing whether two graphs have the same number of connected components. See SL for more information.

Every binary matrix with constant row and column sums is the incidence matrix of a regular uniform block design. Also, each configuration has a corresponding biregular bipartite graph known as its incidence or Levi graph.

Permutation graphs are a special case of circle graphs, comparability graphs, the complements of comparability graphs, and trapezoid graphs. The subclasses of the permutation graphs include the bipartite permutation graphs (characterized by ) and the cographs.

The administrator not only has to decide whether or not to take the applicant but, if so, also has to assign her permanently to one of the jobs. The objective is to find an assignment where the sum of qualifications is as big as possible. This problem is identical to finding a maximum-weight matching in an edge-weighted bipartite graph where the n nodes of one side arrive online in random order. Thus, it is a special case of the online bipartite matching problem.

Phadke demonstrated to Ellis that it is incorrect to classify desires, preferences, demands and commands as Beliefs- rational or irrational. In order to incorporate the human motives in the A-B-C theory, he coined the more comprehensive term for B- Bedrock of Biosocial Forces. In the later years, he reinterpreted letter 'B' more accurately as bipartite belief system. It includes the detection of binary message which the client signals to himself as well as the bipartite belief system, implied in that message.Digiuseppe,Raymond(2007).

In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. The triangle-free graphs with the most edges for their vertices are balanced complete bipartite graphs. Many triangle-free graphs are not bipartite, for example any cycle graph Cn for odd n > 3\.

The POU domain is a bipartite domain composed of two subunits separated by a non-conserved region of 15-55 aa. The N-terminal subunit is known as the POU-specific (POUs) domain (), while the C-terminal subunit is a homeobox domain (). 3D structures of complexes including both POU subdomains bound to DNA are available. Both subdomains contain the structural motif 'helix-turn-helix', which directly associates with the two components of bipartite DNA binding sites, and both are required for high affinity sequence-specific DNA-binding.

If the algorithm terminates without finding an odd cycle in this way, then it must have found a proper coloring, and can safely conclude that the graph is bipartite.. For the intersection graphs of n line segments or other simple shapes in the Euclidean plane, it is possible to test whether the graph is bipartite and return either a two- coloring or an odd cycle in time O(n\log n), even though the graph itself may have up to O\left(n^2\right) edges. .

A graph is a parity graph if and only if every component of its split decomposition is either a complete graph or a bipartite graph. Based on this characterization, it is possible to test whether a given graph is a parity graph in linear time. The same characterization also leads to generalizations of some graph optimization algorithms from bipartite graphs to parity graphs. For instance, using the split decomposition, it is possible to find the weighted maximum independent set of a parity graph in polynomial time..

Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching. The Tutte theorem provides a characterization for arbitrary graphs. A perfect matching is a spanning 1-regular subgraph, a.k.a. a 1-factor.

In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.

Every regular bipartite graph is also biregular. Every edge-transitive graph (disallowing graphs with isolated vertices) that is not also vertex-transitive must be biregular.. In particular every edge-transitive graph is either regular or biregular.

Powerlifting Rio 2016 Paralympic Games – Qualification Guide Two further quota places per event will be awarded by the Bipartite Commission. The following is a summary of the qualification places at the 2016 Summer Paralympic Powerlifting event.

The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge-transitive non-bipartite graphs with odd vertex degrees.. Lauri and Scapelleto credit this construction to Mark Watkins.

Then a perusal of the elastics may suggest a re-ordering on one side or the other with the elastics slightly less tense. After some iteration this procedure reveals a cluster structure in the bipartite graph.

According to the Kővári–Sós–Turán theorem, every -vertex -biclique-free graph has edges, significantly fewer than a dense graph would have.. This work concerns the number of edges in biclique-free bipartite graphs, but a standard application of the probabilistic method transfers the same bound to arbitrary graphs. Conversely, if a graph family is defined by forbidden subgraphs or closed under the operation of taking subgraphs, and does not include dense graphs of arbitrarily large size, it must be -biclique-free for some , for otherwise it would include large dense complete bipartite graphs. As a lower bound, conjectured that every maximal -biclique-free bipartite graph (one to which no more edges can be added without creating a -biclique) has at least edges, where and are the numbers of vertices on each side of its bipartition..

In this scenario, any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the hidden influence's propagation speed. Quantum experiments with three or more parties can, nonetheless, disprove all such non-local hidden variable models.

By Turán's theorem, the n-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible.

5.. Crown graphs themselves are shown to require long (possibly longest) representations among bipartite graphs M. Glen, S. Kitaev, and A. Pyatkin. On the representation number of a crown graph, Discr. Appl. Math. 244, 2018, 89–93..

Balance does not imply bipartiteness. Let H be the hypergraph: > { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: > { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}.

A split graph, partitioned into a clique and an independent set. In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . had a more general definition, in which the graphs they called split graphs also included bipartite graphs (that is, graphs that be partitioned into two independent sets) and the complements of bipartite graphs (that is, graphs that can be partitioned into two cliques).

Geminiviridae are transmitted by white flies, scientifically known as Bamesia tabacci. The genus begomoviridae generally comprises bipartite (two components, namely DNA- ‘A’ and ‘B’) genome, which replicates via rolling circle (RCR) model with the help of few viral and several host factors. Mungbean yellow mosaic India virus (MYMIV) is a representative of the genus begomovirus / Begomoviridae, which is prevalent in northern part of Indian subcontinent causing yellow mosaic disease (YMD). MYMIV possesses bipartite ssDNA genomes named as DNA-A and DNA-B, both being ~2.7 kb in size.

A bipartite graph is a simple graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2. In a complete bipartite graph, the vertex set is the union of two disjoint sets, W and X, so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X.

Zarankiewicz wrote works on cut-points in connected spaces, on conformal mappings, on complex functions and number theory, and triangular numbers. The Zarankiewicz problem is named after Zarankiewicz. This problem asks, for a given size of (0,1)-matrix, how many matrix entries must be set equal to 1 in order to guarantee that the matrix contains at least one a × b submatrix is made up only of 1's. An equivalent formulation in extremal graph theory asks for the maximum number of edges in a bipartite graph with no complete bipartite subgraph Ka,b.

Levin and Roy tested the graph for cut points and failed to find any with their search starting with corporations with large boards. In a study of clustering in the graph, Levin and Roy demonstrated the use of a bipartite graph with corporations listed on one side and directors with multiple seats on the other. The clusters become evident in a physical model using elastic bands and paper clips. The directors and corporations are listed arbitrarily to begin and the elastic bands placed as edges of the bipartite graph.

Similar problems describe the degree sequences of simple bipartite graphs or the degree sequences of simple directed graphs. The first problem is the so-called bipartite realization problem. The second is known as the digraph realization problem. The problem of constructing a solution for the graph realization problem with the additional constraint that each such solution comes with the same probability was shown to have a polynomial-time approximation scheme for the degree sequences of regular graphs by Cooper, Martin, and Greenhill.. The general problem is still unsolved.

Bipartite graphs are extensively used in modern coding theory, especially to decode codewords received from the channel. Factor graphs and Tanner graphs are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.. A factor graph is a closely related belief network used for probabilistic decoding of LDPC and turbo codes., p. 686.

In this way, a labeling of this type for the complete graph corresponds to a partition of its edges into complete bipartite graphs, with the lengths of the labels corresponding to the number of graphs in the partition.

This problem was shown to be in FPTAS for regular sequences by Catherine Greenhill (for regular bipartite graphs with a forbidden 1-factor) and for half-regular sequences by Erdős et al. The general problem is still unsolved.

However, not all planar series-parallel graphs are outerplanar. The complete bipartite graph K2,3 is planar and series-parallel but not outerplanar. On the other hand, the complete graph K4 is planar but neither series-parallel nor outerplanar.

There is a constant-factor approximation algorithm for interval graphs and for bipartite graphs. The interval graph case remains NP- hard. It is the case arising in Supowit's original application in VLSI design, and also has applications in scheduling.

In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.

Many of these results have analogues for balanced bipartite graphs, in which the vertex degrees are compared to the number of vertices on a single side of the bipartition rather than the number of vertices in the whole graph.

A hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X. To see that this sense is stronger than 2-colorability, let H be a hypergraph on the vertices {1, 2, 3, 4} with the following hyperedges: > { {1,2,3} , {1,2,4} , {1,3,4} , {2,3,4} } This H is 2-colorable, for example by the partition X = {1,2} and Y = {3,4}. However, it is not exactly-2-colorable, since every set X with one element has an empty intersection with one hyperedge, and every set X with two or more elements has an intersection of size 2 or more with at least two hyperedges. Every bipartite graph G = (X+Y, E) is exactly-2-colorable. Hall's marriage theorem has been generalized from bipartite graphs to exactly-2-colorable hypergraphs; see Hall-type theorems for hypergraphs.

One reason for interest in the computational complexity of the permanent is that it provides an example of a problem where constructing a single solution can be done efficiently but where counting all solutions is hard. As Papadimitriou writes in his book Computational Complexity: Specifically, computing the permanent (shown to be difficult by Valiant's results) is closely connected with finding a perfect matching in a bipartite graph, which is solvable in polynomial time by the Hopcroft–Karp algorithm.John E. Hopcroft, Richard M. Karp: An n^{5/2} Algorithm for Maximum Matchings in Bipartite Graphs. SIAM J. Comput. 2(4), 225–231 (1973) For a bipartite graph with 2n vertices partitioned into two parts with n vertices each, the number of perfect matchings equals the permanent of its biadjacency matrix and the square of the number of perfect matchings is equal to the permanent of its adjacency matrix.

Chile qualified one competitors in the men's single event, Robinson Mendez. This spot was a result of a Bipartite Commission Invitation place. Chile qualified two competitors in the women's singles event. Francisca Mardones earned her spot via the standard qualification process.

Sigma Aldrich anti-MIPOL1 polyclonal antibody produced in rabbit (HPA002893). Retrieved 27 July 2020. A bipartite nuclear localization signal is predicted at position 128 – 143, which is highly conserved in mammalian orthologs (see Fig.2.), indicating possible localization in the nucleus.

A partition into exactly n-1 complete bipartite graphs is easy to obtain: just order the vertices, and for each vertex except the last, form a star connecting it to all later vertices in the ordering. Other partitions are also possible.

For instance, the complete bipartite graph K3,3 may be formed using the first operation to form a triangle and then applying the second operation to subdivide each edge of the triangle and connect each subdivision point with the opposite triangle vertex.

Esat Hilmi Bayindir earned the right to go. Because of a process to help countries that are underrepresented at the Paralympics, Mehmet Çekiç was invited to participate. This process is called Bipartite Invitation. Esat Hilmi Bayindir was in a traffic accident.

Due to these differences and the perception of an advantage that international qualifications have over HKDSE in university admission, there has been a considerable amount of concern over the emergence of a bipartite education system, based on wealth instead of merit.

The rostrum is bipartite, forming a double point. The two points are widely separated, each broad at the base and tapering forwards. The postorbital spine is laterally expanded and the antorbital spine is absent. Many areas of the carapace have tubercles.

The National Resident Matching Program applies graph matching methods to solve this problem for U.S. medical student job-seekers and hospital residency jobs.. The Dulmage–Mendelsohn decomposition is a structural decomposition of bipartite graphs that is useful in finding maximum matchings..

Deciding whether a graph admits a perfect matching can be done in polynomial time, using any algorithm for finding a maximum cardinality matching. However, counting the number of perfect matchings, even in bipartite graphs, is #P-complete. This is because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix. A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm.

A graph is line perfect if and only if each of its biconnected components is a bipartite graph, the complete graph K_4, or a triangular book K_{1,1,n}. Because these three types of biconnected component are all perfect graphs themselves, every line perfect graph is itself perfect. By similar reasoning, every line perfect graph is a parity graph, a Meyniel graph, and a perfectly orderable graph. Line perfect graphs generalize the bipartite graphs, and share with them the properties that the maximum matching and minimum vertex cover have the same size, and that the chromatic index equals the maximum degree.

An optimal drawing of , with 18 crossings In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II. A drawing method found by Kazimierz Zarankiewicz has been conjectured to give the correct answer for every complete bipartite graph, and the statement that this is true has come to be known as the Zarankiewicz crossing number conjecture. The conjecture remains open, with only some special cases solved.

A rook polynomial is a special case of one kind of matching polynomial, which is the generating function of the number of k-edge matchings in a graph. The rook polynomial Rm,n(x) corresponds to the complete bipartite graph Km,n . The rook polynomial of a general board B ⊆ Bm,n corresponds to the bipartite graph with left vertices v1, v2, ..., vm and right vertices w1, w2, ..., wn and an edge viwj whenever the square (i, j) is allowed, i.e., belongs to B. Thus, the theory of rook polynomials is, in a sense, contained in that of matching polynomials.

Let G = (U ∪ V, E) be a bipartite graph without isolated nodes; all edges are of the form {u, v} ∈ E with u ∈ U and v ∈ V. Then {U, V} is both a vertex 2-coloring and a domatic partition of size 2; the sets U and V are independent dominating sets. The chromatic number of G is exactly 2; there is no vertex 1-coloring. The domatic number of G is at least 2. It is possible that there is a larger domatic partition; for example, the complete bipartite graph Kn,n for any n ≥ 2 has domatic number n.

There exist planar non- Hamiltonian graphs in which all faces have five or eight sides. For these graphs, Grinberg's formula taken modulo three is always satisfied by any partition of the faces into two subsets, preventing the application of his theorem to proving non-Hamiltonicity in this case . It is not possible to use Grinberg's theorem to find counterexamples to Barnette's conjecture, that every cubic bipartite polyhedral graph is Hamiltonian. Every cubic bipartite polyhedral graph has a partition of the faces into two subsets satisfying Grinberg's theorem, regardless of whether it also has a Hamiltonian cycle .

The homomorphism problem with a fixed graph H on the right side of each instance is also called the H-coloring problem. When H is the complete graph Kk, this is the graph k-coloring problem, which is solvable in polynomial time for k = 0, 1, 2, and NP-complete otherwise. In particular, K2-colorability of a graph G is equivalent to G being bipartite, which can be tested in linear time. More generally, whenever H is a bipartite graph, H-colorability is equivalent to K2-colorability (or K0 / K1-colorability when H is empty/edgeless), hence equally easy to decide.

In the illustration, every odd cycle in the graph contains the blue (the bottommost) vertices, so removing those vertices kills all odd cycles and leaves a bipartite graph. The edge bipartization problem is the algorithmic problem of deleting as few edges as possible to make a graph bipartite and is also an important problem in graph modification algorithmics. This problem is also fixed-parameter tractable, and can be solved in time O\left(2^k m^2\right), where k is the number of edges to delete and m is the number of edges in the input graph.

In a complete graph or complete bipartite graph, every cut is a split. In a cycle graph of length four, the partition of the vertices given by 2-coloring the cycle is a nontrivial split, but for cycles of any longer length there are no nontrivial splits. A bridge of a graph that is not 2-edge-connected corresponds to a split, with each side of the split formed by the vertices on one side of the bridge. The cut-set of the split is just the single bridge edge, which is a special case of a complete bipartite subgraph.

2-satisfiability has also been applied to problems of recognizing undirected graphs that can be partitioned into an independent set and a small number of complete bipartite subgraphs,. inferring business relationships among autonomous subsystems of the internet,. and reconstruction of evolutionary trees..

Bolded amino acids are highly conserved even in orthologs as distant as Cnidarians. Other bracketed regions show conserved protein family regions identified such as COG1196 and COG 4372. Exon boundaries are highlighted in blue. Bipartite nuclear localization signal is highlighted in blue.

Hereditary maximal-clique irreducible graphs include triangle-free graphs, bipartite graphs, and interval graphs. Cographs can be characterized as graphs in which every maximal clique intersects every maximal independent set, and in which the same property is true in all induced subgraphs.

The contraction of the degree of saturation forms the name of the algorithm. DSatur is a heuristic graph colouring algorithm, yet produces exact results for bipartite, cycle, and wheel graphs. DSatur has also been referred to as saturation LF in the literature.

CCDC37 has a predicted nuclear localization via Reinhard's methodA. Reinhardt and T. Hubbard, Nucleic Acids Res. 26, 2230, 1998 (reliability 94.1%) using a bipartite nuclear localization signal peptide starting at amino acid 155: KRQMFLLQYALDVKRRE.Dingwall C, Robbins J, Dilworth SM, Roberts B, Richardson WD (Sep 1988).

When phrased as a graph theory problem, the assignment problem can be extended from bipartite graphs to arbitrary graphs. The corresponding problem, of finding a matching in a weighted graph where the sum of weights is maximized, is called the maximum weight matching problem.

Similar theorems describe the degree sequences of simple directed graphs, simple directed graphs with loops, and simple bipartite graphs . The first problem is characterized by the Fulkerson–Chen–Anstee theorem. The latter two cases, which are equivalent, are characterized by the Gale–Ryser theorem.

Adcock retired from active duty in 1947, but was recalled to serve as the U.S. Chairman of the Bipartite Control Office, part of the Military Government in Germany.Marquis Who's Who, Inc. Who Was Who in American History, the Military. Chicago: Marquis Who's Who, 1975.

Marite Kirikova (2002) Information Systems Development: Advances in Methodologies, Components, and Management. p.194. states: The original theory of conceptual graphs was introduced by Sowa (Sowa, 1984 ). A conceptual graph is a finite, connected, bipartite graph. It includes notions of concepts, relations, and actors...

Similar theorems describe the degree sequences of simple graphs, simple directed graphs with loops, and simple bipartite graphs. The first problem is characterized by the Erdős–Gallai theorem. The latter two cases, which are equivalent see Berger, are characterized by the Gale–Ryser theorem.

Part II includes material on repeating-key and bipartite systems and periodic ciphers. Part II, volume II includes the Zendian Problem, a practical exercise in traffic analysis and cryptanalysis. For the Aegean Park Press edition, Wayne G. Barker added programs for the TRS-80.

Kőnig's theorem is equivalent to numerous other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem.

France qualified four competitors in the men's single event, Frederic Cattaneo, Stephane Houdet, Michael Jeremiasz and Nicolas Peifer. France qualified two players in the women's singles event. Charlotte Famin qualified via the standard route. Charlotte Famin earned her slot via a Bipartite Commission Invitation place.

In the case of systems composed of m > 2 subsystems, the classification of quantum-entangled states is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states.

The Herschel graph is a planar graph: it can be drawn in the plane with none of its edges crossing. It is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph. It is a bipartite graph: its vertices can be separated into two subsets of five and six vertices respectively, such that every edge has an endpoint in each subset (the red and blue subsets in the picture). As with any bipartite graph, the Herschel graph is a perfect graph : the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph.

According to computational experiments, the auction algorithm is generally inferior to other state-of- the-art algorithms for the all destinations shortest path problem, but is very fast for problems with few destinations (substantially more than one and substantially less than the total number of nodes); see the article by Bertsekas, Pallottino, and Scutella, Polynomial Auction Algorithms for Shortest Paths. Auction algorithms for shortest hyperpath problems have been defined by De Leone and Pretolani in 1998. This is also a parallel auction algorithm for weighted bipartite matching, described by E. Jason Riedy in 2004."The Parallel Auction Algorithm for Weighted Bipartite Matching", E. Jason Riedy, UC Berkeley, February 2004, .

Any graph with a nonempty set of edges requires at least two colors; if G and H are not 1-colorable, that is, they both contain an edge, then their product also contains an edge, and is hence not 1-colorable either. In particular, the conjecture is true when G or H is a bipartite graph, since then its chromatic number is either 1 or 2. Similarly, if two graphs G and H are not 2-colorable, that is, not bipartite, then both contain a cycle of odd length. Since the product of two odd cycle graphs contains an odd cycle, the product G × H is not 2-colorable either.

Suppose that a quantum system consist of Nparticles. A bipartition of the system is a partition which divide the system into two parts A and B, containing k and l particles respectively with k+l=N. Bipartite entanglement entropy is defined with respect to this bipartition.

Clerodendrum golden mosaic China virus (ClGMCNV) is a bipartite Begomovirus isolated from flowering plants in the Clerodendrum genus. The virus causes yellow mosaic disease in various plant species, including Nicotiana, Petunia, Solanum, and Capsicum species. It is associated with a mosaic disease known as 'Dancing Flame'.

However, not every (3,6)-sparse graph is planar. Similarly, outerplanar graphs are (2,3)-sparse and planar bipartite graphs are (2,4)-sparse. Streinu and Theran show that testing (k,l)-sparsity may be performed in polynomial time when k and l are integers and 0 ≤ l < 2k.

The above example is a special case of the following general theorem: > G is a bipartite graph if-and-only-if MP(G) = FMP(G) if-and-only-if all > corners of FMP(G) have only integer coordinates. This theorem can be proved in several ways.

As of April 2015, crossing numbers are known for very few graph families. In particular, except for a few initial cases, the crossing number of complete graphs, bipartite complete graphs, and products of cycles all remain unknown, although there has been some progress on lower bounds.

In coding theory, a Tanner graph, named after Michael Tanner, is a bipartite graph used to state constraints or equations which specify error correcting codes. In coding theory, Tanner graphs are used to construct longer codes from smaller ones. Both encoders and decoders employ these graphs extensively.

Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form K4 (the tetrahedron) or K1,1,n (a book of one or more triangles all sharing a common edge). Every line perfect graph is itself perfect.

In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of graphs with perfect matchings. It is a generalization of Hall's marriage theorem from bipartite to arbitrary graphs. It is a special case of the Tutte–Berge formula.

Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem and Wagner's theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K4 or the complete bipartite graph K2,3.; ; , Proposition 7.3.1, p. 117; .

In the multipartite case there is no simple necessary and sufficient condition for separability like the one given by the PPT criterion for the 2\otimes2 and 2\otimes3 cases. However, many separability criteria used in the bipartite setting can be generalized to the multipartite case.

The most divergent of the Kainji languages are Reshe, Laru and Lopa, which may form a branch together. Subclassification of the other branches is not yet clear. A bipartite division between East Kainji and West Kainji is no longer maintained, with West Kainji now being paraphyletic.

A bipartite graph with 4 vertices on each side, 13 edges, and no K3,3 subgraph, and an equivalent set of 13 points in a 4 × 4 grid, showing that z(4; 3) ≥ 13\. The number z(n, 2) asks for the maximum number of edges in a bipartite graph with n vertices on each side that has no 4-cycle (its girth is six or more). Thus, z(2, 2) = 3 (achieved by a three-edge path), and z(3, 2) = 6 (a hexagon). In his original formulation of the problem, Zarankiewicz asked for the values of z(n; 3) for n = 4, 5, and 6. The answers were supplied soon afterwards by Wacław Sierpiński: z(4; 3) = 13, z(5; 3) = 20, and z(6; 3) = 26.. The case of z(4; 3) is relatively simple: a 13-edge bipartite graph with four vertices on each side of the bipartition, and no K3,3 subgraph, may be obtained by adding one of the long diagonals to the graph of a cube.

However, when k is even, this is not possible, because in this case the folded cube is a bipartite graph with equal numbers of vertices on each side of the bipartition, very different from the nearly two-to-one ratio for the bipartition of a complete binary tree..

The size of an adult shell varies between 25 mm and 40 mm. (Freely translated from the original Latin description) The whitish shell is elongated and subturrited. The shell contains about 8 rather flat whorls (the apex is lacking). Just below the suture it shows a bipartite girdle.

Depending upon the requirements of the investigation, there are several techniques available to define a DNA- recognition domain that will confer the specificity of a ZFP-based transcription factor. Three phage display strategies have been described, involving either parallel, sequential or bipartite selection of the constituent zinc fingers.

Female marsupials have paired uteri and cervices. Most eutherian (placental) mammal species have a single cervix and single, bipartite or bicornuate uterus. Lagomorphs, rodents, aardvarks and hyraxes have a duplex uterus and two cervices. Lagomorphs and rodents share many morphological characteristics and are grouped together in the clade Glires.

The half graph has a unique perfect matching. This is straightforward to see by induction: u_n must be matched to its only neighbor, v_n, and the remaining vertices form another half graph. More strongly, every bipartite graph with a unique perfect matching is a subgraph of a half graph.

The dessin d'enfant corresponding to the sextic monomial p(x) = x6. The Chebyshev polynomials and the corresponding dessins d'enfants, alternately-colored path graphs. The simplest bipartite graphs are the trees. Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface.

Historically, Nicaragua had a two-party system, with varying two dominant political parties. The 2006 general election could have marked the end of the bipartite scheme, as the anti-Sandinista forces split into two major political alliances: the Nicaraguan Liberal Alliance (ALN) and the Constitutionalist Liberal Party (PLC).

More generally, it has been proven that every complete bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound. Closing the gap between this lower bound and the upper bound remains an open problem.

It is possible to test in polynomial time whether a given binary matroid is bipartite.. However, any algorithm that tests whether a given matroid is Eulerian, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time..

The most famous open problem about list edge-coloring is probably the list coloring conjecture. :ch′(G) = χ′(G). This conjecture has a fuzzy origin; overview its history. The Dinitz conjecture, proven by , is the special case of the list coloring conjecture for the complete bipartite graphs Kn,n.

Owens, P. J. "Bipartite Cubic Graphs and a Shortness Exponent." Discrete Math. 44, 327-330, 1983. The first Ellingham-Horton graph was published by Ellingham in 1981 and was of order 78.Ellingham, M. N. "Non- Hamiltonian 3-Connected Cubic Partite Graphs."Research Report No. 28, Dept. of Math.

Egervary's theorem can be extended, using a similar argument, to graphs that have both edge-weights w and vertex-weights b: > In any edge-weighted vertex-weighted bipartite graph, the maximum w-weight > of a b-matching equals the minimum b-weight of vertices in a w-vertex-cover.

The church is orientated, built from wooden framework, bipartite. The chancel is surrounded by three walls. The nave is wider, in the shape of a square. In the seventeenth-century, the church was surrounded with picturesque soboty (wooden undercut supported by pillars), during which a bell tower was constructed.

As observes, every comparability graph that is neither complete nor bipartite has a skew partition. The complement of any interval graph is a comparability graph. The comparability relation is called an interval order. Interval graphs are exactly the graphs that are chordal and that have comparability graph complements.

Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour. And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for finding such a matching is possible than for irregular graphs: by selecting every other edge of an Euler tour, one may find a 2-factor, which in this case must be a collection of cycles, each of even length, with each vertex of the graph appearing in exactly one cycle. By selecting every other edge again in these cycles, one obtains a perfect matching in linear time.

Although the truth of Barnette's conjecture remains unknown, computational experiments have shown that there is no counterexample with fewer than 86 vertices.; . If Barnette's conjecture turns out to be false, then it can be shown to be NP-complete to test whether a bipartite cubic polyhedron is Hamiltonian.. If a planar graph is bipartite and cubic but only of connectivity 2, then it may be non-Hamiltonian, and it is NP-complete to test Hamiltonicity for these graphs.. Another result was obtained by : if the dual graph can be vertex-colored with colors blue, red and green such that every red-green cycle contains a vertex of degree 4, then the primal graph is Hamiltonian.

It is important in the context of cutting- plane methods for integer programming to be able to describe accurately the facets of polytopes that have vertices corresponding to the solutions of combinatorial optimization problems. Often, these problems have solutions that can be described by binary vectors, and the corresponding polytopes have vertex coordinates that are all zero or one. As an example, consider the Birkhoff polytope, the set of n × n matrices that can be formed from convex combinations of permutation matrices. Equivalently, its vertices can be thought of as describing all perfect matchings in a complete bipartite graph, and a linear optimization problem on this polytope can be interpreted as a bipartite minimum weight perfect matching problem.

Gowers's construction for the lower bound of Szemerédi's regularity lemma first introduced a weaker version of this lemma, restricted to bipartite graphs, in order to prove Szemerédi's theorem,. and in he proved the full lemma.. Extensions of the regularity method to hypergraphs were obtained by Rödl and his collaborators... and Gowers... János Komlós, Gábor Sárközy and Endre Szemerédi later (in 1997) proved in the blow-up lemma that the regular pairs in Szemerédi regularity lemma behave like complete bipartite graphs under the correct conditions. The lemma allowed for deeper exploration into the nature of embeddings of large sparse graphs into dense graphs. The first constructive version was provided by Alon, Duke, Lefmann, Rödl and Yuster.

Hamiltonicity can be expressed in MSO2 by the existence of a set of edges that forms a connected 2-regular graph on all the vertices, with connectivity expressed as above and 2-regularity expressed as the incidence of two but not three distinct edges at each vertex. However, Hamiltonicity is not expressible in MSO1, because MSO1 is not capable of distinguishing complete bipartite graphs with equal numbers of vertices on each side of the bipartition (which are Hamiltonian) from unbalanced complete bipartite graphs (which are not).; , Corollary 7.24, pp. 126–127. Although not part of the definition of MSO2, orientations of undirected graphs can be represented by a technique involving Trémaux trees.

The list chromatic index of a graph is the smallest number with the property that, no matter how one chooses lists of colors for the edges, as long as each edge has at least colors in its list, then a coloring is guaranteed to be possible. Thus, the list chromatic index is always at least as large as the chromatic index. The Dinitz conjecture on the completion of partial Latin squares may be rephrased as the statement that the list edge chromatic number of the complete bipartite graph equals its edge chromatic number, . resolved the conjecture by proving, more generally, that in every bipartite graph the chromatic index and list chromatic index are equal.

The problem of finding the smallest odd cycle transversal, or equivalently the largest bipartite induced subgraph, is also called odd cycle transversal, and abbreviated as OCT. It is NP-hard, as a special case of the problem of finding the largest induced subgraph with a hereditary property (as the property of being bipartite is hereditary). All such problems for nontrivial properties are NP-hard. The equivalence between the odd cycle transversal and vertex cover problems has been used to develop fixed-parameter tractable algorithms for odd cycle transversal, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k.

For graphs that are not bipartite, the minimum vertex cover may be larger than the maximum matching. Moreover, the two problems are very different in complexity: maximum matchings can be found in polynomial time for any graph, while minimum vertex cover is NP-complete. The complement of a vertex cover in any graph is an independent set, so a minimum vertex cover is complementary to a maximum independent set; finding maximum independent sets is another NP-complete problem. The equivalence between matching and covering articulated in Kőnig's theorem allows minimum vertex covers and maximum independent sets to be computed in polynomial time for bipartite graphs, despite the NP-completeness of these problems for more general graph families.

Kőnig's theorem is named after the Hungarian mathematician Dénes Kőnig. Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching,In a poster displayed at the 1998 International Congress of Mathematicians in Berlin and again at the Bled'07 International Conference on Graph Theory, Harald Gropp has pointed out that the same result already appears in the language of configurations in the 1894 thesis of Ernst Steinitz. and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree. – the latter statement is known as Kőnig's line coloring theorem.

The complement of A forms a vertex cover in G with the same cardinality as this matching. This connection to bipartite matching allows the width of any partial order to be computed in polynomial time. More precisely, n-element partial orders of width k can be recognized in time O(kn2) .

Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.

The protein encoded by this gene is a subunit of a bipartite UDP-N-acetylglucosamine transferase. It heterodimerizes with asparagine-linked glycosylation 14 (ALG14) homolog to form a functional UDP-GlcNAc glycosyltransferase that catalyzes the second sugar addition of the highly conserved oligosaccharide precursor in endoplasmic reticulum N-linked glycosylation.

If the chromatic number of a graph is uncountable, then the graph necessarily contains as a subgraph a half graph on the natural numbers. This half graph, in turn, contains every complete bipartite graph in which one side of the bipartition is finite and the other side is countably infinite.

Sometimes this theorem is stated with the additional constraint b_1\geq\cdots\geq b_n. This condition is not necessary, because the labels of vertices of one partite set in a bipartite graph can be switched arbitrarily. In 1962 Ford and Fulkerson gave a different but equivalent formulation for the theorem.

U.S.-Brazil Cooperative Research: Problems on Random Graphs (Structures) and Set Systems: NSG GRANT 0072064 Kohayakawa has an Erdős number of 1.Celina Miraglia Herrera – My Erdős numberHe wrote "The size of the largest bipartite subgraphs", on Discrete Mathematics with Erdős and Gyárfás He was awarded the 2018 Fulkerson Prize.

More generally the snarks are defined as the graphs that, like the Petersen graph, are bridgeless, 3-regular, and of class 2. According to the theorem of , every bipartite regular graph has a 1-factorization. The theorem was stated earlier in terms of projective configurations and was proven by Ernst Steinitz.

Cographs are exactly the graphs with clique-width at most 2. Every distance-hereditary graph has clique-width at most 3. However, the clique-width of unit interval graphs is unbounded (based on their grid structure). Similarly, the clique-width of bipartite permutation graphs is unbounded (based on similar grid structure).

However, attribute Kőnig's theorem itself to a later paper of Kőnig (1931). According to , Kőnig attributed the idea of studying matchings in bipartite graphs to his father, mathematician Gyula Kőnig. In Hungarian, Kőnig's name has a double acute accent, but his theorem is usually spelled in German characters, with an umlaut.

The United States qualified two competitors in the men's single event as a result of Bipartite Commission Invitation places. The players invited were Steve Baldwin and Jon Rydberg. The United States qualified four players in the women's single event. Emmy Kaiser and Dana Mathewson qualified via the standard qualification route.

Process systems are a dualistic phenomenon of change/no-change or form/transform and as such, are well-suited to being modeled by the bipartite Petri nets modeling system and in particular, process-class dualistic Petri nets where processes can be simulated in real time and space and studied hierarchically.

In Christian theological anthropology, bipartite refers to the view that a human being is composed of two distinct components, material and immaterial, body and soul. The two parts were created interdependent and in harmony, though corrupted through sin. Alternative theological views of human composition include tripartite and unitary (or monistic) views.

Such placentas are described as bilobed/bilobular/bipartite, trilobed/trilobular/tripartite, and so on. If there is a clearly discernible main lobe and auxiliary lobe, the latter is called a succenturiate placenta. Sometimes the blood vessels connecting the lobes get in the way of fetal presentation during labor, which is called vasa previa.

Let H be the hypergraph: > { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge.

In the case r=2, the hypergraph becomes a bipartite graph, and the conjecture becomes \tau(H) \leq u(H). This is known to be true by Kőnig's theorem. In the case r=3, the conjecture has been proved by Ron Aharoni. The proof uses the Aharoni-Haxell theorem for matching in hypergraphs.

The biclique partition problem takes as input an arbitrary undirected graph, and asks for a partition of its edges into a minimum number of complete bipartite graphs. It is NP-hard, but fixed-parameter tractable. The best approximation algorithm known for the problem has an approximation ratio of O(n/\log n).

The columella is arcuate, terminating in a bipartite tooth at the base. (Further description by G.W. Tryon) There are 5 whorls. Above the shoulder angle there are two shallow spiral furrows. Between this and the peripheral carina there are 4, of equal breadth to the elevated interspaces; and on the base about 12.

These viruses have segmented, bipartite genomes that are linear, positive-sense, single-stranded RNA (1). These genomes are about 5300 nucleotides in length (1). They have a methylated cap at the 5'-end whose sequence type is m7GpppA (1). The genome also codes for non-structural proteins as well as structural proteins (1).

Given a graph, deciding whether it is the square of another graph is NP-complete. . Moreover, it is NP-complete to determine whether a graph is a kth power of another graph, for a given number k ≥ 2, or whether it is a kth power of a bipartite graph, for k > 2..

The virions are icosahedral, non enveloped and ~25 nanometers in diameter. They contain two structural proteins. The genome is bipartite, unique among ssDNA viruses, with two linear segments of ~6 and 6.5 kilobases. These segments and the complementary strands are that are packaged separately giving rise to 4 different types of full particles.

The smallest cubic graphs with crossing numbers 1–8 and 11 are known . The smallest 1-crossing cubic graph is the complete bipartite graph , with 6 vertices. The smallest 2-crossing cubic graph is the Petersen graph, with 10 vertices. The smallest 3-crossing cubic graph is the Heawood graph, with 14 vertices.

3-dimensional matchings. (a) Input T. (b)–(c) Solutions. In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs. Finding a largest 3-dimensional matching is a well-known NP-hard problem in computational complexity theory.

Because K1,5 has maximum degree five, the number of colors guaranteed for it by the Hajnal–Szemerédi theorem is six, achieved by giving each vertex a distinct color. Another interesting phenomenon is exhibited by a different complete bipartite graph, K2n + 1,2n + 1. This graph has an equitable 2-coloring, given by its bipartition.

A blossom is a factor-critical subgraph of a larger graph. Blossoms play a key role in Jack Edmonds' algorithms for maximum matching and minimum weight perfect matching in non-bipartite graphs.. In polyhedral combinatorics, factor-critical graphs play an important role in describing facets of the matching polytope of a given graph.

This protein is also a component of other multisubunit complexes e.g. thyroid hormone receptor-(TR-) associated proteins which interact with TR and facilitate TR function on DNA templates in conjunction with initiation factors and cofactors. This protein contains a bipartite nuclear localization signal. This gene is known to escape chromosome X-inactivation.

The one exceptional case is L(K4,4), which shares its parameters with the Shrikhande graph. When both sides of the bipartition have the same number of vertices, these graphs are again strongly regular., Theorem 8.7, p. 79\. Harary credits this characterization of line graphs of complete bipartite graphs to Moon and Hoffman.

Köln: Rüdiger Köppe, 2010. Like many other Berber varieties, the Figuig Berber dialects use bipartite verbal negation. The preverbal negator is ul (locally un, il); the postverbal negator is ša (Igli, Mazzer) / šay (Figuig, Iche, Moghrar) / iš (Boussemghoun, Ain Chair), with both the latter two appearing as allomorphs in Tiout.Kossmann, op. cit.

Kaitlyn Verfuerth qualified via a Bipartite Commission Invitation place. Russia had qualified two player in the women's singles event, Ludmila Bubnova and Viktoriia Lvova. Following their suspension, one spot was re-allocated by the IPC to the United States' Shelby Baron. The United States qualified three players in the quad singles event.

In graph theory, the Gallai–Edmonds decomposition is a partition of the vertices of a graph into subsets satisfying certain properties. It is a generalization of Dulmage–Mendelsohn decomposition from bipartite graphs to general graphs. It was proved independently by Tibor Gallai and Jack Edmonds. It can be found using the blossom algorithm.

Iofinova and Ivanov proved in 1985 the existence of five and only five semi-symmetric cubic bipartite graphs whose automorphism groups act primitively on each partition. The smallest has 110 vertices. The others have 126, 182, 506 and 990. The 126-vertex Iofinova-Ivanov graph is also known as the Tutte 12-cage.

To say that P is necessary and sufficient for Q is to say two things: # that P is necessary for Q, P \Leftarrow Q, and that P is sufficient for Q, P \Rightarrow Q. # equivalently, it may be understood to say that P and Q is necessary for the other, P \Rightarrow Q \land Q \Rightarrow P, which can also be stated as each is sufficient for or implies the other. One may summarize any, and thus all, of these cases by the statement "P if and only if Q", which is denoted by P \Leftrightarrow Q, whereas cases tell us that P \Leftrightarrow Q is identical to P \Rightarrow Q \land Q \Rightarrow P. For example, in graph theory a graph G is called bipartite if it is possible to assign to each of its vertices the color black or white in such a way that every edge of G has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and conversely.

In their original paper Reed et al. showed how to make a graph bipartite by deleting at most k vertices in time O(3k kmn). Later, a simpler algorithm was given, also using iterative compression, by Lokshstanov, Saurabh and Sikdar.. In order to compress a deletion set of size to a deletion set of size , their algorithm tests all of the partitions of into three subsets: the subset of that belongs to the new deletion set, and the two subsets of that belong to the two sides of the bipartite graph that remains after deleting . Once these three sets have been selected, the remaining vertices of a deletion set (if it exists) can be found from them by applying a max-flow min-cut algorithm.

The relation "contains" can be represented by a bipartite graph. The vertices of the graph are divided into two disjoint sets, one representing the subsets in and another representing the elements in X. If a subset contains an element, an edge connects the corresponding vertices in the graph. In the graph representation, an exact cover is a selection of vertices corresponding to subsets such that each vertex corresponding to an element is connected to exactly one selected vertex. For example, the relation "contains" in the detailed example above can be represented by a bipartite graph with 6+7 = 13 vertices: 300px Again, the subcollection = {B, D, F} is an exact cover, since each element is contained in exactly one selected subset, i.e.

In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particular p. 181.. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942.. The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure.

Consider the complete bipartite graph G = K2,4, having six vertices A, B, W, X, Y, Z such that A and B are each connected to all of W, X, Y, and Z, and no other vertices are connected. As a bipartite graph, G has usual chromatic number 2: one may color A and B in one color and W, X, Y, Z in another and no two adjacent vertices will have the same color. On the other hand, G has list-chromatic number larger than 2, as the following construction shows: assign to A and B the lists {red, blue} and {green, black}. Assign to the other four vertices the lists {red, green}, {red, black}, {blue, green}, and {blue, black}.

Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing. The study of crossing numbers originated in Turán's brick factory problem, in which Pál Turán asked for a factory plan that minimized the number of crossings between tracks connecting brick kilns to storage sites. Mathematically, this problem can be formalized as asking for the crossing number of a complete bipartite graph, The same problem arose independently in sociology at approximately the same time, in connection with the construction of sociograms. Turán's conjectured formula for the crossing numbers of complete bipartite graphs remains unproven, as does an analogous formula for the complete graphs.

The theory of perfect graphs developed from a 1958 result of Tibor Gallai that in modern language can be interpreted as stating that the complement of a bipartite graph is perfect; this result can also be viewed as a simple equivalent of Kőnig's theorem, a much earlier result relating matchings and vertex covers in bipartite graphs. The first use of the phrase "perfect graph" appears to be in a 1963 paper of Claude Berge, after whom Berge graphs are named. In this paper he unified Gallai's result with several similar results by defining perfect graphs, and he conjectured the equivalence of the perfect graph and Berge graph definitions; his conjecture was proved in 2002 as the strong perfect graph theorem.

In the mentioned method, it is claimed and proved that finding a flow value of k in G between s and t is equal to finding a feasible schedule for flight set F with at most k crews. Another version of airline scheduling is finding the minimum needed crews to perform all the flights. In order to find an answer to this problem, a bipartite graph is created where each flight has a copy in set A and set B. If the same plane can perform flight j after flight i, i∈A is connected to j∈B. A matching in induces a schedule for F and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews.

The bipartite graph shown in the above illustration has 14 vertices; a matching with six edges is shown in blue, and a vertex cover with six vertices is shown in red. There can be no smaller vertex cover, because any vertex cover has to include at least one endpoint of each matched edge (as well as of every other edge), so this is a minimum vertex cover. Similarly, there can be no larger matching, because any matched edge has to include at least one endpoint in the vertex cover, so this is a maximum matching. Kőnig's theorem states that the equality between the sizes of the matching and the cover (in this example, both numbers are six) applies more generally to any bipartite graph.

Egerváry's theorem says: > In any edge-weighted bipartite graph, the maximum w-weight of a matching > equals the smallest number of vertices in a w-vertex-cover. The maximum w-weight of a fractional matching is given by the LP: > Maximize w · x Subject to: x ≥ 0E __________ AG · x ≤ 1V. And the minimum number of vertices in a fractional w-vertex-cover is given by the dual LP: > Minimize 1V · y Subject to: y ≥ 0V __________ AGT · y ≥ w. As in the proof of Konig's theorem, the LP duality theorem implies that the optimal values are equal (for any graph), and the fact that the graph is bipartite implies that these programs have optimal solutions in which all values are integers.

Parity graphs include the distance-hereditary graphs, in which every two induced paths between the same two vertices have the same length. They also include the bipartite graphs, which may be characterized analogously as the graphs in which every two paths (not necessarily induced paths) between the same two vertices have the same parity, and the line perfect graphs, a generalization of the bipartite graphs. Every parity graph is a Meyniel graph, a graph in which every odd cycle of length five or more has two chords. For, in a parity graph, any long odd cycle can be partitioned into two paths of different parities, neither of which is a single edge, and at least one chord is needed to prevent these from both being induced paths.

Cryptomonad flagella are inserted parallel to one another, and are covered by bipartite hairs called mastigonemes, formed within the endoplasmic reticulum and transported to the cell surface. Small scales may also be present on the flagella and cell body. The mitochondria have flat cristae, and mitosis is open; sexual reproduction has also been reported.

MIPOL1 contains two coiled-coil domains in its C-terminus at positions 107 – 212 and 253 – 435 (shown in Fig.1). A bipartite nuclear localization signal is predicted at position 128 – 143. Fig.3. Annotated conceptual translation of MIPOL1 isoform 1 showing the most important features. The underlined parts represent the coiled-coil domains.

In the other direction, if a bipartite graph with 14 edges has four vertices on each side, then two vertices on each side must have degree four. Removing these four vertices and their 12 incident edges leaves a nonempty set of edges, any of which together with the four removed vertices forms a K3,3 subgraph.

The pre-mRNA of this subunit is edited at one position. The R/G editing site is located in exon 13 between the M3 and M4 regions. Editing results in a codon change from an arginine (AGA) to a glycine (GGA). The location of editing corresponds to a bipartite ligand interaction domain of the receptor.

In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges. It has chromatic number 2, chromatic index 3, diameter 5, radius 5 and girth 6.Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput.

Kondo H, Maeda T, Shirako Y, Tamada T 2006. Orchid fleck virus is a rhabdovirus with an unusual bipartite genome. J Gen Virol 87:2413–2421. Much is still not known about OFV including how exactly and why vector mites travel from orchid to orchid, and more host species of flowers are being discovered annually.

Hull was an undergraduate at Hampshire College. He earned a master's degree and Ph.D. in mathematics at the University of Rhode Island. His 1997 dissertation, Some Problems in List Coloring Bipartite Graphs, involved graph coloring, and was supervised by Nancy Eaton. Prior to his appointment at Western New England, Hull taught at Merrimack College.

The pre-mRNA of this subunit is edited at one position. The R/G editing site is located in exon 13 between the M3 to M4 region. Editing results in a codon change from an Arginine (AGA) to a Glycine (GGA). The location of editing corresponds to a bipartite ligand interaction domain of the receptor.

A permutation set with positive entries is equivalent to a perfect matching in the positivity graph. A perfect matching in a bipartite graph can be found in polynomial time, e.g. using any algorithm for maximum cardinality matching. Kőnig's theorem is equivalent to the following: > The positivity graph of any bistochastic matrix admits a perfect matching.

The barycentric subdivision subdivides each edge of the graph. This is a special subdivision, as it always results in a bipartite graph. This procedure can be repeated, so that the nth barycentric subdivision is the barycentric subdivision of the n−1th barycentric subdivision of the graph. The second such subdivision is always a simple graph.

W is involved in spermatogenesis, telomere associated functions in sperm and is found in spermatogenic cells. It is characterized by the extension of the N-terminal tail. subH2B participates in regulation of spermiogenesis and is found in non-nucleosomal particle in the subacrosome of spermatozoa. This variant has a bipartite nuclear localization signal. H2B.

Bede, Saint. The Ecclesiastical History of the English People: The Greater Chronicle; Bede's Letter to Egbert. Oxford University Press, 1994. Simon Keynes suggests Egbert's foundation of a 'bipartite' kingdom is crucial as it stretched across southern England, and it created a working alliance between the West Saxon dynasty and the rulers of the Mercians.

The length of the fusiform shell is 53 mm and its diameter 16 mm. The shell is rather narrow with a very long spire and short siphonal canal. It is corded with larger and smaller riblets and raised lines. The shell is very slightly angulated on each whorl by a somewhat larger rib, which is occasionally bipartite.

Given a finite colored directed bipartite graph with n vertices V = V_0 \cup V_1, and V colored with colors from 1 to m, is there a choice function selecting a single out-going edge from each vertex of V_0, such that the resulting subgraph has the property that in each cycle the largest occurring color is even.

The recognition of intrinsic immunity as a potent anti-viral defense mechanism is a recent discovery and is not yet discussed in most immunology courses or texts. Though the extent of protection intrinsic immunity affords is still unknown, it is possible that intrinsic immunity may eventually be considered a third branch of the traditionally bipartite immune system.

For bipartite pure entangled states that can be transformed in this way with unit probability, the respective Schmidt coefficients are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. Others have shown how quantum catalytic behaviour arises under a probabilistic approach via stochastic dominance with respect to the convolution of measures.

The exact cover problem is NP-complete This book is a classic, developing the theory, then cataloguing many NP-Complete problems. and is one of Karp's 21 NP- complete problems. The exact cover problem is a kind of constraint satisfaction problem. An exact cover problem can be represented by an incidence matrix or a bipartite graph.

The maximum size bicluster is equivalent to maximum edge biclique in bipartite graph. In the complex case, the element in matrix A is used to compute the quality of a given bicluster and solve the more restricted version of the problem. It requires either large computational effort or the use of lossy heuristics to short-circuit the calculation.

This article details the qualifying phase for Table tennis at the 2016 Summer Paralympics. The competition at these Games will comprise a total of 276 athletes coming from their respective NPCs. 31 (20 male, 11 female) will be selected by the Bipartite Commission. A further 55 male and 50 female quota places were reserved for continental competition.

They involve two separated lovers, one of who sends the other a message, and thus are designed to evoke the ' ('feeling of love'). And they adhere to a bipartite structure in which the first half charts the journey the messenger is to follow, while the second describes the messenger's destination, the recipient and the message itself.

The Petersen graph is a cubic graph. The complete bipartite graph K_{3,3} is an example of a bicubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.

This approximation ratio is close to best possible, as under standard complexity-theoretic assumptions a ratio of (1-\epsilon)\log n cannot be achieved in polynomial time for any \epsilon>0 . The latter hardness of approximation still holds for instances restricted to subcubic graphs , and even to bipartite subcubic graphs as shown in Hartung's PhD thesis .

An NPC can be allocated a maximum of thirty-four male qualification slots and a maximum of twenty-eight female qualification slots for a maximum total of sixty-two qualification slots per NPC. Exceptions may be granted through the Bipartite Invitation Commission Allocation method.Qualification Guide - Rio 2016, published by paralympics.org, the Official Website of the IPC.

Most involve two separated lovers, one of whom sends the other a message, and thus are designed to evoke the śṛṅgāra rasa ('feeling of love'). They tend to adhere to a bipartite structure in which the first half charts the journey the messenger is to follow, while the second describes the messenger’s destination, the recipient and the message itself.

A flow diagram of the VIC CipherThe VIC cipher was a pencil and paper cipher used by the Soviet spy Reino Häyhänen, codenamed "VICTOR". If the cipher were to be given a modern technical name, it would be known as a "straddling bipartite monoalphabetic substitution superenciphered by modified double transposition." David Kahn. "Number One From Moscow". 1993\.

The hnRNP proteins have distinct nucleic acid binding properties. The protein encoded by this gene contains a RNA binding domain and scaffold-associated region (SAR)-specific bipartite DNA-binding domain. This protein is also thought to be involved in the packaging of hnRNA into large ribonucleoprotein complexes. During apoptosis, this protein is cleaved in a caspase-dependent way.

In a perfect claw-free graph, the neighborhood of any vertex forms the complement of a bipartite graph. It is possible to color perfect claw-free graphs, or to find maximum cliques in them, in polynomial time., pp. 135–136. In general, it is NP-hard to find the largest clique in a claw-free graph.

The fastest algorithm for trapezoid order recognition was proposed by McConnell and Spinrad in 1994, with a running time of O(n^2). The process reduces the interval dimension 2 question to a problem of covering an associated bipartite graph by chain graphs (graphs with no induced 2K2).Golumbic, Martin Charles, and Ann N. Trenk. Tolerance Graphs.

DYRK1B is a member of the DYRK family of protein kinases. DYRK1B contains a bipartite nuclear localization signal and is found mainly in muscle and testis. The protein is proposed to be involved in the regulation of nuclear functions. Three isoforms of DYRK1B have been identified differing in the presence of two alternatively spliced exons within the catalytic domain.

On the top left a Latin square, on the bottom left the relative proper n-edge coloring. On the top right a Latin transversal and on the bottom right the relative rainbow matching. Rainbow matchings are of particular interest given their connection to transversals of Latin squares. Denote by Kn,n the complete bipartite graph on n+n vertices.

The colors of Toruń are white and blue in the horizontal arrangement, white top, blue bottom, equal in size. The flag of the city of Toruń is a bipartite sheet. The upper field is white, the lower field is blue. If the flag is hung vertically, the upper edge of the flag must be on the left.

A graph with two nontrivial strong splits (top) and its split decomposition (bottom). The three quotient graphs are a star (left), a prime graph (center), and a complete graph (right). In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph. A graph is prime if it has no splits.

Mantel's theorem states that any n-vertex undirected graph with at least n2/4 edges, and no multiple edges or self-loops, either contains a triangle or it is the complete bipartite graph Kn/2,n/2. This theorem can be strengthened: any undirected Hamiltonian graph with at least n2/4 edges is either pancyclic or Kn/2,n/2. There exist n-vertex Hamiltonian directed graphs with n(n + 1)/2 − 3 edges that are not pancyclic, but every Hamiltonian directed graph with at least n(n + 1)/2 − 1 edges is pancyclic. Additionally, every n-vertex strongly connected directed graph in which each vertex has degree at least n (counting incoming and outgoing edges together) is either pancyclic or it is a complete bipartite directed graph..

Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be realizable in that geometry), or as a type of abstract incidence geometry. In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the girth of the corresponding bipartite graph (the Levi graph of the configuration) must be at least six.

If an infinite graph G has a normal spanning tree, so does every connected graph minor of G. It follows from this that the graphs that have normal spanning trees have a characterization by forbidden minors. One of the two classes of forbidden minors consists of bipartite graphs in which one side of the bipartition is countable, the other side is uncountable, and every vertex has infinite degree. The other class of forbidden minors consists of certain graphs derived from Aronszajn trees.. The details of this characterization depend on the choice of set-theoretic axiomatization used to formalize mathematics. In particular, in models of set theory for which Martin's axiom is true and the continuum hypothesis is false, the class of bipartite graphs in this characterization can be replaced by a single forbidden minor.

WSMV is a flexible, rod-shaped virus composed of a positive-sense single-strand RNA genome approximately 8.5 to 12 kilobases in length, and can be either mono- or bipartite. The RNA serves as both the genome and viral messenger. The genomic RNA (or its segments) is translated into polyprotein(s) which are transformed by virus-encoded proteases into functional products.

The VCX gene cluster is polymorphic in terms of copy number; different individuals may have a different number of VCX genes. VCX/Y genes encode small and highly charged proteins of unknown function. The presence of a putative bipartite nuclear localization signal suggests that VCX/Y members are nuclear proteins. This gene contains 10 repeats of the 30-bp unit.

R. intermedia is found in the bottom sludge of freshwater bodies in Canada, Chile, Argentina, Australia, New Zealand, Malaysia, Russia, and central Europe. Raphidiophrys have bipartite scales are a defining characteristic among species. Differences in type and size of scales are used to differentiate amongst the members of this genus. The genus Raphidiophrys was discovered in 1867 by W. Archer.

In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges. It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3. Its girth is 10 and there are exactly 168 cycles of length 10 in it. There are also 168 cycles of length 12.

After the Horton graph, a number of smaller counterexamples to the Tutte conjecture were found. Among them are a 92 vertex graph by Horton published in 1982, a 78 vertex graph by Owens published in 1983, and the two Ellingham-Horton graphs (54 and 78 vertices).Horton, J. D. "On Two-Factors of Bipartite Regular Graphs." Discrete Math. 41, 35-41, 1982.

Finding a radio coloring with a given (or minimum) span is NP-complete, even when restricted to planar graphs, split graphs, or the complements of bipartite graphs.. However it is solvable in polynomial time for trees and cographs.. For arbitrary graphs, it can be solved in singly-exponential time, significantly faster than a brute-force search through all possible colorings...

The utility graph K3,3 is a circulant graph. It is the (3,4)-cage, the smallest triangle-free cubic graph. Like all other complete bipartite graphs, it is a well-covered graph, meaning that every maximal independent set has the same size. In this graph, the only two maximal independent sets are the two sides of the bipartition, and obviously they are equal.

The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement. In classical information theory , the Shannon entropy, is associated to a probability distribution,p_1, \cdots, p_n, in the following way: : H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i.

Afar, Agaw, Oromo and Somali). Leo Reinisch subsequently grouped Beja with Saho-Afar, Somali and Oromo in a Lowland Cushitic sub- phylum, representing one half of a two-fold partition of Cushitic. Moreno (1940) proposed a bipartite classification of Beja similar to that of Reinisch, but lumped Beja with both Lowland Cushitic and Central Cushitic. Around the same period, Enrico Cerulli (c.

Consider a matchmaking agency with a pool of men and women. Given the preferences of the candidates, the agency constructs a bipartite graph where there is an edge between a man and a woman iff they are compatible. The ultimate goal of the agency is to create as many compatible couples as possible, i.e., find a maximum-cardinality matching in this graph.

In monotremes (egg-laying mammals) such as the platypus, the uterus is duplex and rather than nurturing the embryo, secretes the shell around the egg. It is essentially identical with the shell gland of birds and reptiles, with which the uterus is homologous. In mammals, the four main forms of the uterus are: duplex, bipartite, bicornuate and simplex.Lewitus, Eric, and Christophe Soligo.

Flowers have salver-form, meaning starting from a narrow tube and suddenly flaring into a flat arrangement of petals. Flowers are white or pale lemon- yellow, orange when fading. Flower tube is about 2 inches long, with 5-9 obliquely obovate petals, about 1/2 as long as the tube. Stigma is club- shaped, thick, and fleshy, bipartite, segments bifid.

Indeed, any interval which is closed on the left is the union of a singular interval and of an interval open on the left, in this order. And similarly for intervals closed on the right. A signal automaton reading a bipartite signal has a special form. Its set of locations can be partitioned into locations for singular interval, and locations for open intervals.

A Hamiltonian path (but not cycle) in the Herschel graph As a bipartite graph that has an odd number of vertices, the Herschel graph does not contain a Hamiltonian cycle (a cycle of edges that passes through each vertex exactly once). For, in any bipartite graph, any cycle must alternate between the vertices on either side of the bipartition, and therefore must contain equal numbers of both types of vertex and must have an even length. Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph. It is the smallest non-Hamiltonian polyhedral graph, whether the size of the graph is measured in terms of its number of vertices, edges, or faces.. There exist other polyhedral graphs with 11 vertices and no Hamiltonian cycles (notably the Goldner–Harary graph.) but none with fewer edges.

Historic American Buildings Survey photograph To accommodate the needs of the congregation, Wright divided the community space from the temple space through a low, middle loggia that could be approached from either side. This was an efficient use of space and kept down on noise between the two main gathering areas: those coming for religious services would be separated via the loggia from those coming for community events. The plan of Wright's design looks back to the bipartite design of his own studio built several blocks away in 1898: with two portions of the building similar in composition and separated by a lower passageway, and one section being larger than the other (the Guggenheim Museum in New York City is another bipartite design). Also for the Temple's architecture, Wright borrowed several attributes from his previous creation, the Larkin Administration Building.

A second graph-theoretic description of the problem is also possible. Once the women have been seated, the possible seating arrangements for the remaining men can be described as perfect matchings in a graph formed by removing a single Hamiltonian cycle from a complete bipartite graph; the graph has edges connecting open seats to men, and the removal of the cycle corresponds to forbidding the men to sit in either of the open seats adjacent to their wives. The problem of counting matchings in a bipartite graph, and therefore a fortiori the problem of computing ménage numbers, can be solved using the permanents of certain 0-1 matrices. In the case of the ménage problem, the matrix arising from this view of the problem is the circulant matrix in which all but two adjacent elements of the generating row equal .

The case k = 2 is trivial: a graph requires more than one color if and only if it has an edge, and that edge is itself a K2 minor. The case k = 3 is also easy: the graphs requiring three colors are the non-bipartite graphs, and every non-bipartite graph has an odd cycle, which can be contracted to a 3-cycle, that is, a K3 minor. In the same paper in which he introduced the conjecture, Hadwiger proved its truth for k ≤ 4\. The graphs with no K4 minor are the series- parallel graphs and their subgraphs. Each graph of this type has a vertex with at most two incident edges; one can 3-color any such graph by removing one such vertex, coloring the remaining graph recursively, and then adding back and coloring the removed vertex.

Sometimes recorded as Berkasova in the literature, however, Berkasovo is the correct transliteration of the name of the Serbian village where the helmets were excavated. The bipartite construction method is usually characterized by a two-part bowl united by a central ridge running from front to back and small cheekpieces, also it lacks a base-ring running around the rim of the bowl. Some examples of the bipartite construction also utilize metal crests, such as in the Intercisa-IV and River Maas examples. The second type of helmet has a quadripartite construction, characterized by a four-piece bowl connected by a central ridge, with two plates (connected by a reinforcing band) on each side of the ridge, and a base-ring uniting the elements of the skull at the rim of the helmet; this type is further characterised by large cheekpieces.

And, a planar graph is bipartite if and only if, in a planar embedding of the graph, all face cycles have even length. Therefore, Barnette's conjecture may be stated in an equivalent form: suppose that a three-dimensional simple convex polyhedron has an even number of edges on each of its faces. Then, according to the conjecture, the graph of the polyhedron has a Hamiltonian cycle.

The optimum key generation problem is to find a minimum set of encryption keys for ensuring secure transmission. As above, the problem can be modeled using a bipartite graph whose minimum biclique edge covers coincide with the solutions to the optimum key generation problem . A different application lies in biology, where minimum biclique edge covers are used in mathematical models of human leukocyte antigen (HLA) serology .

We consider several bipartite graphs with Y = {1, 2} and X = {A, B; a, b, c}. The Meshulam condition trivially holds for the empty set. It holds for subsets of size 1 iff the neighbor-graph of each vertex in Y is non-empty (so it requires at least one explosion to destroy), which is easy to check. It remains to check the subset Y itself.

However, on non-bipartite lattices, RVB liquid phases possessing topological order and fractionalized spinons also appear. The discovery of topological order in quantum dimer models (more than a decade after the models were introduced) has led to new interest in these models. Classical dimer models have been studied previously in statistical physics, in particular by P. W. Kasteleyn (1961) and M. E. Fisher (1961).

As proved, the maximum independent set of any ideal polyhedron (the largest possible subset of non-adjacent vertices) must have at most half of the vertices of the polyhedron. It can have exactly half only when the vertices can be partitioned into two equal-size independent sets, so that the graph of the polyhedron is a balanced bipartite graph, as it is for an ideal cube.

SH3 domain-binding protein 4 is a protein that in humans is encoded by the SH3BP4 gene. This gene encodes a protein with 3 Asn-Pro-Phe (NPF) motifs, an SH3 domain, a PXXP motif, a bipartite nuclear targeting signal, and a tyrosine phosphorylation site. This protein is involved in cargo-specific control of clathrin-mediated endocytosis, specifically controlling the internalization of a specific protein receptor.

Perpendicular Gothic, it has an octagonal two-stage tower, with corbelled shafts at the angles to the upper stage flanking bipartite belfrey louvred lights; the parapet has battlements and truncated pinnacles are located at the angles. Ardeer church currently has 2 ministers; Rev John Lafferty and Rev David Hebenton who are shared between Ardeer and their linked church, Livingstone Parish Church, just a short distance away.

There are various ways of approximating this number from first principles. Suppose there are a given number N of water molecules. The oxygen atoms form a bipartite lattice: they can be divided into two sets, with all the neighbors of an oxygen atom from one set lying in the other set. Focus attention on the oxygen atoms in one set: there are N/2 of them.

Extremal combinatorics studies extremal questions on set systems. The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on 2n vertices is a complete bipartite graph Kn,n. Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate.

The overall organization of MTA3 protein domains is similar to the other two family members with a BAH (Bromo-Adjacent Homology), an ELM2 (egl-27 and MTA1 homology), a SANT (SWI, ADA2, N-CoR, TFIIIB-B), a GATA-like zinc finger, and one predicted bipartite nuclear localization signal (NLS). The SH3 motif of Mta3 allows it to interact with Fyn and Grb2 – both SH3 containing signaling proteins.

G4 A gear graph, denoted Gn is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges. Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. Gear graphs are also known as cogwheels and bipartite wheels.

A petri net (PN) is a bipartite graph of places and transitions used as a model for QDI circuits. Transitions in the petri net represent voltage transitions on nodes in the circuit. Places represent the partial states between transitions. A token inside a place acts as a program counter identifying the current state of the system and multiple tokens may exist in a petri net simultaneously.

One may also consider playing either Geography game on an undirected graph (that is, the edges can be traversed in both directions). Fraenkel, Scheinerman, and Ullman show that undirected vertex geography can be solved in polynomial time, whereas undirected edge geography is PSPACE- complete, even for planar graphs with maximum degree 3. If the graph is bipartite, then Undirected Edge Geography is solvable in polynomial time.

A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle.

A generalization of PageRank for the case of ranking two interacting groups of objects was described in. In applications it may be necessary to model systems having objects of two kinds where a weighted relation is defined on object pairs. This leads to considering bipartite graphs. For such graphs two related positive or nonnegative irreducible matrices corresponding to vertex partition sets can be defined.

Nonsteroidal estrogens prevalently exist in our environment and have both positive and negative effects on our daily life. But as a possible way to get access to neurodegenerative disease treatment, scientists have developed multiple ways to screen these estrogens and select the ones that have less side effects. Bipartite recombinant yeast system and dual fluorescence report system are designed to screen these potential chemicals.

It says that, if deg(G) ≥ n/2, then the graph admits a Hamiltonian cycle; this implies that it admits a perfect matching. The factor n/2 is tight, since the complete bipartite graph on (n/2-1, n/2+1) vertices has degree n/2-1 but does not admit a perfect matching. The results described below aim to extend these results from graphs to hypergraphs.

It was led by Pierre Dupong, and also included three other Ministers. The head of state, Grand Duchess Charlotte, also escaped from Luxembourg after the occupation. The government was bipartite, including two members from both the Party of the Right (PD) and the Socialist Workers' Party (LSAP). The government was located in 27 Wilton Crescent in Belgravia, London which now serves as the Luxembourgish Embassy in London.

Notable among the Neoplatonic commentators in this regard are Alexander, Themistius, Philoponus, and Simplicius. The part dealing with method argues that the procedure of collection in the Sophist is superseded by the use of paradigms in the Statesman, and that bipartite division in the Sophist is replaced by multipartite division in service of a method similar to the method of negation employed in the Parmenides.

Just as Content and Consciousness has a bipartite structure, he similarly divided Brainstorms into two sections. He would later collect several essays on content in The Intentional Stance and synthesize his views on consciousness into a unified theory in Consciousness Explained. These volumes respectively form the most extensive development of his views. In chapter 5 of Consciousness Explained Dennett describes his multiple drafts model of consciousness.

Fandango is a lively couples dance originating from Portugal and Spain, usually in triple metre, traditionally accompanied by guitars, castanets, or hand-clapping ("palmas" in Spanish). Fandango can both be sung and danced. Sung fandango is usually bipartite: it has an instrumental introduction followed by "variaciones". Sung fandango usually follows the structure of "cante" that consist of four or five octosyllabic verses (coplas) or musical phrases (tercios).

The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored.; ). However, nonplanar triangle-free graphs may require many more than three colors.

Macroneuropteris scheuchzeri specimen clearly showing the hair- like structures on the leaves. The foliage of the Macroneuropteris species consists of very large frond-like leaves that are bipartite (divided in two) near the base, forming two large bipinnately compound parts (see illustration). These compound fronds can be as large as several meters. In Macroneuropteris, each individual leaflet or pinnule of the compound frond were also notably large.

This is because, on one hand, a 3-coloring of G is the same as a homomorphism G → K3, as explained below. On the other hand, every subgraph of G trivially admits a homomorphism into G, implying K3 → G. This also means that K3 is the core of any such graph G. Similarly, every bipartite graph that has at least one edge is equivalent to K2.

In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete bipartite graph K3,3., p. 77; .

If G contains a clique of size k, then at least k colors are needed to color that clique; in other words, the chromatic number is at least the clique number: : \chi(G) \ge \omega(G). For perfect graphs this bound is tight. Finding cliques is known as the clique problem. The 2-colorable graphs are exactly the bipartite graphs, including trees and forests.

Because these graphs are bipartite and have Hamiltonian paths, their maximum independent sets have a number of vertices that is equal to half of the number of vertices in the whole graph, rounded up to the nearest integer., p.6. The diameter of a Fibonacci cube of order n is n, and its radius is n/2 (again, rounded up to the nearest integer)., p.9.

The matching complex of a complete bipartite graph Km,n is known as a chessboard complex. It is the clique graph of the complement graph of a rook's graph,. and each of its simplices represents a placement of rooks on an m × n chess board such that no two of the rooks attack each other. When m = n ± 1, the chessboard complex forms a pseudo-manifold.

Megabirnaviridae is a family of double-stranded RNA viruses with one genus Megabirnavirus which infects fungi. The group name derives from member's bipartite dsRNA genome and mega that is greater genome size (16 kbp) than families Birnaviridae (6 kbp) and Picobirnaviridae (4 kbp). There is only one species in this family: the type species Rosellinia necatrix megabirnavirus 1. Diseases associated with this family include: reduced host virulence.

Deciding whether the metric dimension of a graph is at most a given integer is NP- complete . It remains NP-complete for bounded-degree planar graphs , split graphs, bipartite graphs and their complements, line graphs of bipartite graphs , unit disk graphs , interval graphs of diameter 2 and permutation graphs of diameter 2 . For any fixed constant k, the graphs of metric dimension at most k can be recognized in polynomial time, by testing all possible k-tuples of vertices, but this algorithm is not fixed-parameter tractable (for the natural parameter k, the solution size). Answering a question posed by , show that the metric dimension decision problem is complete for the parameterized complexity class W[2], implying that a time bound of the form nO(k) as achieved by this naive algorithm is likely optimal and that a fixed-parameter tractable algorithm (for the parameterization by k) is unlikely to exist.

Trees are a special case of bipartite graphs, and testing whether a tree is well-covered can be handled as a much simpler special case of the characterization of well- covered bipartite graphs: if is a tree with more than two vertices, it is well-covered if and only if each non-leaf node of the tree is adjacent to exactly one leaf. The same characterization applies to graphs that are locally tree-like, in the sense that low-diameter neighborhoods of every vertex are acyclic: if a graph has girth eight or more (so that, for every vertex , the subgraph of vertices within distance three of is acyclic) then it is well- covered if and only if every vertex of degree greater than one has exactly one neighbor of degree one.; , Theorem 4.1. A closely related but more complex characterization applies to well-covered graphs of girth five or more.

It is possible to test whether a graph is bipartite, and to return either a two-coloring (if it is bipartite) or an odd cycle (if it is not) in linear time, using depth-first search. The main idea is to assign to each vertex the color that differs from the color of its parent in the depth-first search forest, assigning colors in a preorder traversal of the depth-first-search forest. This will necessarily provide a two-coloring of the spanning forest consisting of the edges connecting vertices to their parents, but it may not properly color some of the non- forest edges. In a depth-first search forest, one of the two endpoints of every non-forest edge is an ancestor of the other endpoint, and when the depth first search discovers an edge of this type it should check that these two vertices have different colors.

For an n \times n knight's graph, the number of vertices is n^2 and the number of edges is 4(n-2)(n-1). A Hamiltonian cycle on the knight's graph is a (closed) knight's tour. A chessboard with an odd number of squares has no tour, because the knight's graph is a bipartite graph and only the bipartite graphs with an even number of vertices can have Hamiltonian cycles. All but finitely many chessboards with an even number of squares have a knight's tour; Schwenk's theorem provides an exact listing of which ones do and which do not.. When it is modified to have toroidal boundary conditions (meaning that a knight is not blocked by the edge of the board, but instead continues onto the opposite edge) the 4\times 4 knight's graph is the same as the four-dimensional hypercube graph.

The bouleuterion in Morgantina is a rectangular building located west of the agora of the city. It was founded during the 3rd century BC, a period of great prosperity for Morgantina, which, from 5th century BC on had acquired a profoundly Hellenic character. The building had a bipartite plan. A walled forecourt led through a stoa to the main entrance, centrally located at the east wall of the auditorium.

Scody Victor, a veteran of the 2012 Summer Paralympics, was 27 years old at the time of the Rio Summer Games. He is classified as S9 because he has a limb deficiency and he uses prosthetic legs to aid in his mobility. The Bipartite Commission invited Victor to partake at the Rio de Janeiro Paralympics. He took part in competitions organised by the Mauritian Swimming Federation to prepare for the Games.

On May 7, 2019, Ohtani played in his first game with the Angels since undergoing Tommy John surgery, batting as a designated hitter against the Detroit Tigers. In a June 13 game against Tampa Bay, Ohtani became the first Japanese-born player to hit for the cycle in MLB history. On September 12, Ohtani's 2019 season prematurely ended after it was revealed that he needed surgery to repair a bipartite patella.

Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian. By Steinitz's theorem, a planar graph represents the edges and vertices of a convex polyhedron if and only if it is polyhedral. A three-dimensional polyhedron has a cubic graph if and only if it is a simple polyhedron.

The problem of determining the bipartite dimension of a graph appears in various contexts of computing. For instance, in computer systems, different users of a system can be allowed or disallowed accessing various resources. In a role-based access control system, a role provides access rights to a set of resources. A user can own multiple roles, and he has permission to access all resources granted by some of his roles.

Further ResearchThey have periplast with longitudinal striations visible in all species. And, Goniomonas is the only Cryptomonad so far examined that does not possess a plastidial complex, and is therefore considered primitive among Cryptophytes. Other Cryptophytes have bipartite tubular flagellar hairs, whereas Goniomonas has solid spike-like flagellar projections. The furrow-gullet system of Goniomonas is located on the anterior of the cell rather than the usual ventral location.

Add a source vertex s and connect it to all the vertices in A′ and add a sink vertex t and connect all vertices inside group B′ to this vertex. The capacity of all the new edges is 1 and their costs is 0. It is proved that there is minimum weight perfect bipartite matching in G if and only if there a minimum cost flow in G′.

Its NRHP nomination states that: > The Saco Mercantile's design embodies the typical characteristics of the > Western Commercial style. The primary facade (south) is organized in a > symmetrical, bipartite arrangement. This facade included a first floor > storefront typical of the period, having a central entrance flanked by > display windows, a lower window panel, and a transom across the top. A > canvas storefront awning is pictured in historic photos of the building.

The Konig property of a hypergraph H is the property that its minimum vertex-cover has the same size as its maximum matching. The Kőnig-Egervary theorem says that all bipartite graphs have the Konig property. The balanced hypergraphs are exactly the hypergraphs H such that every partial subhypergraph of H has the Konig property (i.e., H has the Konig property even upon deleting any number of hyperedges and vertices).

Tubby protein homolog is a protein that in humans is encoded by the TUB gene. This gene encodes a member of the Tubby family of bipartite transcription factors. The encoded protein may play a role in obesity and sensorineural degradation. The crystal structure has been determined for a similar protein in mouse, and it functions as a membrane-bound transcription regulator that translocates to the nucleus in response to phosphoinositide hydrolysis.

Every Halin graph is a Hamiltonian graph, and every edge of the graph belongs to a Hamiltonian cycle. Moreover, any Halin graph remains Hamiltonian after deletion of any vertex. Because every tree without vertices of degree 2 contains two leaves that share the same parent, every Halin graph contains a triangle. In particular, it is not possible for a Halin graph to be a triangle-free graph nor a bipartite graph.

The Birkhoff polytope Bn (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph K_{n,n}) is the convex polytope in RN (where N = n2) whose points are the doubly stochastic matrices, i.e., the matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff.

Artifacts and material from the early and late Bronze Ages have been found. Some of the finds are Iron Age, while some date to the original Neolithic age settlement and others to a later occupation of the site in Iron Age. Pottery remains have been found in both houses. One large vessel found in Yoxie was very similar to a plain Bipartite Urn, possibly used for storing barley.

This gene encodes a protein which specifically interacts with translin, a DNA-binding protein that binds consensus sequences at breakpoint junctions of chromosomal translocations. The encoded protein contains bipartite nuclear targeting sequences that may provide nuclear transport for translin, which lacks any nuclear targeting motifs. Both TSNAX and translin form the C3PO complex which facilitates endonucleolytic cleavage of the passenger strand during microRNA loading into the RNA-induced silencing complex (RISC).

A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition (in fact, regularity is not required for this property to hold). For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other.

It was conjectured by Lovász and Plummer that the number of perfect matchings contained in a cubic, bridgeless graph is exponential in the number of the vertices of the graph .. The conjecture was first proven for bipartite, cubic, bridgeless graphs by , later for planar, cubic, bridgeless graphs by . The general case was settled by , where it was shown that every cubic, bridgeless graph contains at least 2^{n/3656} perfect matchings.

In computer science, Luby transform codes (LT codes) are the first class of practical fountain codes that are near-optimal erasure correcting codes. They were invented by Michael Luby in 1998 and published in 2002.M.Luby, "LT Codes", The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Like some other fountain codes, LT codes depend on sparse bipartite graphs to trade reception overhead for encoding and decoding speed.

Chlorotic and necrotic flecks, spots, and/or ringspots, as well as yellow flecks or spots are all symptoms of an OFV infection. Studies have also shown that OFV may prevent the propagation of other viruses in an already OFV- infected plant.Peng DW, Zheng GH, Zheng ZZ, Tong QX, Ming YL 2013 Orchid fleck virus: an unclassified bipartite, negative-sense RNA plant virus. Archives of Virology.158(2):313-323.

It is NP-complete to determine whether A(G) ≤ 3. showed that the decision variant of the problem is NP-complete even when G is a bipartite graph. demonstrated that every proper vertex coloring of a chordal graph is also an acyclic coloring. Since chordal graphs can be optimally colored in O(n + m) time, the same is also true for acyclic coloring on that class of graphs.

Genome arrangement of dsRNA1 and dsRNA2 of human picobirnavirus The genome is linear, bipartite, and composed of double- stranded RNA. It includes a segment 1 which is 2.2–2.7 kilobases (kb) in length and a segment 2 which is 1.2–1.9 kb in length. The genome codes for three to four proteins. The capsid protein gene is encoded by the second open reading frame of the larger genomic segment 1.

More generally, let q be a positive integer, and let G be the complete bipartite graph Kq,qq. Let the available colors be represented by the q2 different two-digit numbers in radix q. On one side of the bipartition, let the q vertices be given sets of colors } in which the first digits are equal to each other, for each of the q possible choices of the first digit i.

A detailed examination of nucleoplasmin identified a sequence with two elements made up of basic amino acids separated by a spacer arm. One of these elements was similar to the SV40 NLS but was not able to direct a protein to the cell nucleus when attached to a non-nuclear reporter protein. Both elements are required. This kind of NLS has become known as a bipartite classical NLS.

This conjecture implies the unique games conjecture. It has also been used to prove strong hardness of approximation results for finding complete bipartite subgraphs. In 2010, Sanjeev Arora, Boaz Barak and David Steurer found a subexponential time approximation algorithm for the unique games problem.. Previously announced at FOCS 2010. In 2018, after a series of papers, a weaker version of the conjecture, called the 2-2 games conjecture, was proven.

In the mathematical field of graph theory, the Horton graph or Horton 96-graph is a 3-regular graph with 96 vertices and 144 edges discovered by Joseph Horton. Published by Bondy and Murty in 1976, it provides a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph is Hamiltonian.Tutte, W. T. "On the 2-Factors of Bicubic Graphs." Discrete Math. 1, 203-208, 1971/72.

In the other direction, by Kuratowski's theorem, a graph that is not planar necessarily contains a subdivision of either or of the complete bipartite graph . The Kelmans–Seymour conjecture refines this theorem by providing a condition under which only one of these two subdivisions, the subdivision of , can be guaranteed to exist. It states that, if a non-planar graph is 5-vertex-connected, then it contains a subdivision of .

Circular plot of the bipartite mouse X chromosome, generated by the Epigenome Browser. Image from The different 3C-style experiments produce data with very different structures and statistical properties. As such, specific analysis packages exist for each experiment type. Hi-C data is often used to analyze genome-wide chromatin organization, such as topologically associating domains (TADs), linearly contiguous regions of the genome that are associated in 3-D space.

Williams came to UCF from Jones High School in Orlando. He suffered several medical setbacks as he suffered a burst appendix as a freshman, then played through knee pain as a sophomore. Early in his junior season, Williams had tests and learned that his knee pain was due to inferior bipartite patella, a rare congenital condition requiring surgery. Williams had the surgery, missing the majority of that season.

In the mathematical field of graph theory, the Folkman graph, named after Jon Folkman, is a bipartite 4-regular graph with 20 vertices and 40 edges. The Folkman graph is Hamiltonian and has chromatic number 2, chromatic index 4, radius 3, diameter 4 and girth 4\. It is also a 4-vertex-connected and 4-edge- connected perfect graph. It has book thickness 3 and queue number 2.

K7, the complete graph with 7 vertices, is a core. Two graphs G and H are homomorphically equivalent if G → H and H → G. The maps are not necessarily surjective nor injective. For instance, the complete bipartite graphs K2,2 and K3,3 are homomorphically equivalent: each map can be defined as taking the left (resp. right) half of the domain graph and mapping to just one vertex in the left (resp.

The Pappus graph The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and 27 edges., p. 28. The Desargues configuration can also be defined in terms of perspective triangles, and the Reye configuration can be defined analogously from two tetrahedra that are in perspective with each other in four different ways, forming a desmic system of tetrahedra.

First, two lists are made that form two nonintersecting partitions: the list of objects and the list of rules. Objects are denoted by circles. Each rule in a mivar network is an extension of productions, hyper-rules with multi-activators or computational procedures. It is proved that from the perspective of further processing, these formalisms are identical and in fact are nodes of the bipartite graph, denoted by rectangles.

This was one of the first entanglement measures constructed specifically for multipartite states. Definition [Schmidt measure]: The minimum of \; \log r, where \; r is the number of terms in an expansion of the state in product basis. This measure is zero if and only if the state is fully product; therefore, it cannot distinguish between truly multipartite entanglement and bipartite entanglement, but it may nevertheless be useful in many contexts.

A maximum matching in a graph is a set of edges that is as large as possible subject to the condition that no two edges share an endpoint. In a bipartite graph with bipartition (U,V), the sets of edges satisfying the condition that no two edges share an endpoint in U are the independent sets of a partition matroid with one block per vertex in U and with each of the numbers d_i equal to one. The sets of edges satisfying the condition that no two edges share an endpoint in V are the independent sets of a second partition matroid. Therefore, the bipartite maximum matching problem can be represented as a matroid intersection of these two matroids.. More generally the matchings of a graph may be represented as an intersection of two matroids if and only if every odd cycle in the graph is a triangle containing two or more degree-two vertices..

In computer science, a property testing algorithm for a decision problem is an algorithm whose query complexity to its input is much smaller than the instance size of the problem. Typically property testing algorithms are used to decide if some mathematical object (such as a graph or a boolean function) has a "global" property, or is "far" from having this property, using only a small number of "local" queries to the object. For example, the following promise problem admits an algorithm whose query complexity is independent of the instance size (for an arbitrary constant ε > 0): :"Given a graph G on n vertices, decide if G is bipartite, or G cannot be made bipartite even after removing an arbitrary subset of at most \epsilon\tbinom n2 edges of G." Property testing algorithms are central to the definition of probabilistically checkable proofs, as a probabilistically checkable proof is essentially a proof that can be verified by a property testing algorithm.

Partition of the edges of the complete graph K_6 into five complete bipartite subgraphs, according to the Graham–Pollak theorem The Graham–Pollak theorem, which Graham published with Henry O. Pollak in two papers in 1971 and 1972, states that if the edges of an n-vertex complete graph are partitioned into complete bipartite subgraphs, then at least n-1 subgraphs are needed. Graham and Pollak provided a simple proof using linear algebra, and despite the combinatorial nature of the statement and despite multiple publications of alternative proofs since their work, all known proofs require linear algebra. Soon after research in quasi-random graphs began with the work of Andrew Thomason, Graham and his coauthors Fan Chung and R. M. Wilson published in 1989 a result that has been called the "fundamental theorem of quasi-random graphs", stating that many different definitions of these graphs are equivalent. Graham's pebbling conjecture, appearing in a 1989 paper by Fan Chung, concerns the pebbling number of Cartesian products of graphs.

Factor-critical graphs must always have an odd number of vertices, and must be 2-edge-connected (that is, they cannot have any bridges).. However, they are not necessarily 2-vertex- connected; the friendship graphs provide a counterexample. It is not possible for a factor-critical graph to be bipartite, because in a bipartite graph with a near-perfect matching, the only vertices that can be deleted to produce a perfectly matchable graph are the ones on the larger side of the bipartition. Every 2-vertex-connected factor-critical graph with edges has at least different near-perfect matchings, and more generally every factor-critical graph with edges and blocks (2-vertex-connected components) has at least different near-perfect matchings. The graphs for which these bounds are tight may be characterized by having odd ear decompositions of a specific form.. Any connected graph may be transformed into a factor-critical graph by contracting sufficiently many of its edges.

If they do not, then the path in the forest from ancestor to descendant, together with the miscolored edge, form an odd cycle, which is returned from the algorithm together with the result that the graph is not bipartite. However, if the algorithm terminates without detecting an odd cycle of this type, then every edge must be properly colored, and the algorithm returns the coloring together with the result that the graph is bipartite.. Alternatively, a similar procedure may be used with breadth-first search in place of depth-first search. Again, each node is given the opposite color to its parent in the search forest, in breadth-first order. If, when a vertex is colored, there exists an edge connecting it to a previously-colored vertex with the same color, then this edge together with the paths in the breadth-first search forest connecting its two endpoints to their lowest common ancestor forms an odd cycle.

Alexandr Covaliov was taking part in his first Summer Paralympic Games and he was the sole male athlete to compete on Moldova's behalf in Rio. He was aged 32 at the time of the Games. He lost his vision in an accident that burned his eyes at the age of 16 and he took up swimming to aid in his recovery. Covaliov qualified for the Paralympics by receiving an invitation from the Bipartite Commission.

The BLMV genome is bipartite containing two segmented regions of linear, positive-sense, single stranded RNA. The entire genome of the virus is 14600 nucleotides long, and the RNA-1 has a partially sequenced region that is 7600 nucleotides long. The virus consists of a naked, icosahedral capsid that is 28 nm in diameter. The genome of the virus codes for both structural and non-structural proteins, and the lipids of this virus are unknown.

A related conjecture of Barnette states that every cubic polyhedral graph in which all faces have six or fewer edges is Hamiltonian. Computational experiments have shown that, if a counterexample exists, it would have to have more than 177 vertices.. The intersection of these two conjectures would be that every bipartite cubic polyhedral graph in which all faces have four or six edges is Hamiltonian. This was proved to be true by .

Raphidiophrys can be distinguished from other heliozoans as being spineless , however this is not true for Raphidiophrys heterophryoidea . Members of this genus are covered in tangential siliceous scales of one or many types including long, narrow scale with sharp points, narrow ellipsoidal and broad oval scales . Bipartite scales with, sometimes-branched, septa are characteristic of Raphidiophrys . Fine structure in scales and size can be used to differentiate amongst species in the genus .

As with any bipartite graph, there are no odd-length cycles, and there are also no cycles of four or six vertices, so the girth of the Gray graph is 8. The simplest oriented surface on which the Gray graph can be embedded has genus 7 . The Gray graph is Hamiltonian and can be constructed from the LCF notation: :[-25,7,-7,13,-13,25]^9.\ As a Hamiltonian cubic graph, it has chromatic index three.

The complexity of the biclustering problem depends on the exact problem formulation, and particularly on the merit function used to evaluate the quality of a given bicluster. However most interesting variants of this problem are NP-complete. NP-complete have two conditions. In the simple case that there is only element a(i,j) either 0 or 1 in the binary matrix A, a bicluster is equal to a biclique in the corresponding bipartite graph.

Restricted Boltzmann machines (RBMs) are often used as a building block for multilayer learning architectures. An RBM can be represented by an undirected bipartite graph consisting of a group of binary hidden variables, a group of visible variables, and edges connecting the hidden and visible nodes. It is a special case of the more general Boltzmann machines with the constraint of no intra-node connections. Each edge in an RBM is associated with a weight.

If M is a binary matroid that is not Eulerian, then it has a unique Eulerian extension, a binary matroid \bar M whose elements are the elements of M together with one additional element e, such that the restriction of \bar M to the elements of M is isomorphic to M. The dual of \bar M is a bipartite matroid formed from the dual of M by adding e to every odd circuit..

The conjecture has been resolved for the case where n is a prime power by using a topological approach. The conjecture has also been resolved for all non-trivial monotone properties on bipartite graphs by . Minor-closed properties have also been shown to be evasive for large n . In the conjecture was generalized to properties of other (non-graph) functions too, conjecturing that any non-trivial monotone function that is weakly symmetric is evasive.

A bilateral treaty (also called a bipartite treaty) is a treaty strictly between two state entities. It is an agreement made by negotiations between two parties, established in writing and signed by representatives of the parties. Treaties can span in substance and complexity, regarding a wide variety of matters, such as territorial boundaries, trade and commerce, political alliances, and more. The agreement is usually then ratifed by the lawmaking authority of each party or organization.

The steps on the north have been replaced by a ramp providing access for people with disabilities. Above the porches in the gable of the transepts are circular windows with heavy circular tracery, above which are trefoil window openings. The nave elevations are characterised by bipartite lancet openings, in pointed archways, separated by buttressing. The 1939 extension of the western, entrance end is distinguishable by squared headed window openings and wider buttress spacing.

Heat map and circular plot visualization of Hi-C data. a. Hi-C interactions among all chromosomes from G401 human kidney cells, as plotted by the my5C software. b. Heat map visualization illustrating the bipartite structure of the mouse X chromosome, as plotted by Hi-Browse. c. Heat map visualization of a 3 Mbp locus (chr4:18000000-21000000), produced by Juicebox, using in-situ Hi-C data from the GM12878 cell line. d.

The Gray graph is edge-transitive and regular, but not vertex- transitive. Edge-transitive graphs include any complete bipartite graph K_{m,n}, and any symmetric graph, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive.

LCF notation is useful in publishing concise descriptions of Hamiltonian cubic graphs, such as the examples below. In addition, some software packages for manipulating graphs include utilities for creating a graph from its LCF notation.e.g. Maple, NetworkX , igraph, and sage. If a graph is represented by LCF notation, it is straightforward to test whether the graph is bipartite: this is true if and only if all of the offsets in the LCF notation are odd..

By Wagner's theorem, the K_{3,3}-minor-free graphs are formed by gluing together copies of planar graphs and the complete graph K_5 along shared edges. The same gluing structure can be used to obtain a Pfaffian orientation for these graphs. Along with K_{3,3}, there are infinitely many minimal non-Pfaffian graphs. For bipartite graphs, it is possible to determine whether a Pfaffian orientation exists, and if so find one, in polynomial time.

In computer science, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem, in which the input is restricted to be a bipartite graph. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case where all edge weights are the same.

The assignment problem seeks to find a matching in a weighted bipartite graph that has maximum weight. Maximum weighted matchings do not have to be stable, but in some applications a maximum weighted matching is better than a stable one. The matching with contracts problem is a generalization of matching problem, in which participants can be matched with different terms of contracts. An important special case of contracts is matching with flexible wages.

It is obvious from the definition that any vertex-cover must be at least as large as any matching (since for every edge in the matching, at least one vertex is needed in the cover). In particular, the minimum vertex cover is at least as large as the maximum matching. Kőnig's theorem states that, in any bipartite graph, the minimum vertex cover and the maximum matching have in fact the same size., Theorem 5.3, p.

In the case of squashed entanglement, its multipartite version can be obtained by simply replacing the mutual information of the bipartite system with its generalization for multipartite systems, i.e. I(A_1 : \ldots : A_N) = S(A_1) + \ldots + S(A_N) - S(A_1 \ldots A_N). However, in the multipartite setting many more parameters are needed to describe the entanglement of the states, and therefore many new entanglement measures have been constructed, especially for pure multipartite states.

Augsburg-Pfersee Ridge Helmet (Intercisa-type), mid-4th century. This example shows the silver and/or gold sheathing found on most Roman ridge helmets (the helmet would have had cheekpieces – now missing). Unlike earlier Roman helmets, the skull of the ridge helmet is constructed from more than one element. Roman ridge helmets can be classified into two types of skull construction: bipartite and quadripartite, also referred to as Intercisa-type and Berkasovo-type, respectively.

Pupils sat the 11+ examination in their last year of primary education and were sent to one of a secondary modern, secondary technical or grammar school depending on their perceived ability. As it transpired, secondary technical schools were never widely implemented and for 20 years there was a virtual bipartite system which saw fierce competition for the available grammar school places, which varied between 15% and 25% of total secondary places, depending on location.

The bipartite selection (Fig 1 (C)) method was proposed by Isalan et al., 2001 as a compromise between the parallel and sequential selection strategies. The first and last 5 bp of the 9 bp target site are selected in parallel and combined to produce a library from which the final ZFP is chosen. In order to keep the library size within reasonable limits, this technique is restricted to randomisation of only the key residues involved in base recognition.

In the balanced assignment problem, both parts of the bipartite graph have the same number of vertices, denoted by n. One of the first polynomial-time algorithms for balanced assignment was the Hungarian algorithm. It is a global algorithm – it is based on improving a matching along augmenting paths (alternating paths between unmatched vertices). Its run-time complexity, when using Fibonacci heaps, is O(mn + n^2\log n), where m is a number of edges.

Although it does not have the second "classical" NLS, pat7, nor the "non-classical" bipartite NLS it is still predicted to be targeted for the nucleus by the NCNN score. This score predicts whether the protein is targeted for the nucleus or the cytoplasm based upon the amino acid sequence. For the FAM214A protein, the NCNN score predicted nuclear localization with 94.1% certainty. Based upon this information, PSORT generates an overall prediction of the protein's subcellular localization.

See the proof of Theorem 10 in : "Since G contains a 3-cycle consisting of one inner vertex and two outer vertices, G is not a bipartite graph." A Halin graph without any 8-cycles. A similar construction allows any single even cycle length to be avoided. More strongly, every Halin graph is almost pancyclic, in the sense that it has cycles of all lengths from 3 to n with the possible exception of a single even length.

The BEST theorem shows that the number of Eulerian circuits in directed graphs can be computed in polynomial time, a problem which is #P-complete for undirected graphs.Brightwell and Winkler, "Note on Counting Eulerian Circuits", CDAM Research Report LSE-CDAM-2004-12, 2004. It is also used in the asymptotic enumeration of Eulerian circuits of complete and complete bipartite graphs.Brendan McKay and Robert W. Robinson, Asymptotic enumeration of eulerian circuits in the complete graph, Combinatorica, 10 (1995), no.

It is an iterative procedure, so the name "auction algorithm" is related to a sales auction, where multiple bids are compared to determine the best offer, with the final sales going to the highest bidders. The original form of the auction algorithm is an iterative method to find the optimal prices and an assignment that maximizes the net benefit in a bipartite graph, the maximum weight matching problem (MWM).M.G. Resende, P.M. Pardalos. "Handbook of optimization in telecommunications", 2006M.

The graph sandwich problem is NP-complete when Π is the property of being a chordal graph, comparability graph, permutation graph, chordal bipartite graph, or chain graph.. It can be solved in polynomial time for split graphs,. threshold graphs, and graphs in which every five vertices contain at most one four- vertex induced path.. The complexity status has also been settled for the H-free graph sandwich problems for each of the four-vertex graphs H..

One can construct a bipartite graph in which the vertices on one side are the sets, the vertices on the other side are the elements, and the edges connect a set to the elements it contains. Then, a transversal is equivalent to a perfect matching in this graph. One can construct a hypergraph in which the vertices are the elements, and the hyperedges are the sets. Then, a transversal is equivalent to a vertex cover in a hypergraph.

This complementary set induces a matching in G. Each vertex of the independent set is adjacent to n vertices of the matching, and each vertex of the matching is adjacent to n − 1 vertices of the independent set. Because of this decomposition, and because odd graphs are not bipartite, they have chromatic number three: the vertices of the maximum independent set can be assigned a single color, and two more colors suffice to color the complementary matching.

It has been suggested that amalgaviruses have evolved via recombination between viruses with double-stranded and negative-strand RNA genomes. Specifically, phylogenetic analyses have shown that the amalgavirus RdRps form a sister clade to the corresponding proteins of partitiviruses (Partitiviridae) which have segmented (bipartite) dsRNA genomes and infect plants, fungi and protists. By contrast, the putative capsid protein of amalgaviruses is homologous to the nucleocapsid proteins of negative-strand RNA viruses of the genera Phlebovirus (Bunyaviridae) and Tenuivirus.

G × H is connected if and only if both factors are connected and at least one factor is nonbipartite. In particular the bipartite double cover of G is connected if and only if G is connected and nonbipartite. The Hedetniemi conjecture, which gave a formula for the chromatic number of a tensor product, was disproved by . The tensor product of graphs equips the category of graphs and graph homomorphisms with the structure of a symmetric closed monoidal category.

The equine uterus is bipartite, meaning the two uterine horns fuse into a relatively large uterine body (resembling a shortened bicornuate uterus or a stretched simplex uterus). Caudal to the uterus is the cervix, about long, which separates the uterus from the vagina. Usually in diameter with longitudinal folds on the interior surface, it can expand to allow the passage of the foal. The vagina of the mare is long, and is quite elastic, allowing it to expand.

In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph G(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it.

In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 is planar.. Bóna introduces the puzzle (in the form of three houses to be connected to three wells) on p. 275, and writes on p. 277 that it "is equivalent to the problem of drawing K3,3 on a plane surface without crossings". This graph is often referred to as the utility graph in reference to the problem;Utility Graph from mathworld.wolfram.

The first aria, "" (In Him you can dare all), depicts a "hunting scene" for bass and strings. Bach plays on the double meaning of the German word , which in the text has the sense "achieve by great exertion", but he expresses the word's literal meaning ("to hunt") by an "outrageous hunting call trill" of the bass. This aria and those following are not da capo arias, but follow the bar form of the poem as bipartite structures.

In graph theory, the Dulmage–Mendelsohn decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and only if they are paired with each other in a perfect matching of the graph. It is named after A. L. Dulmage and Nathan Mendelsohn, who published it in 1958. A generalization to any graph is the Edmonds–Gallai decomposition, using the Blossom algorithm.

The two components of the genome have very distinct molecular evolutionary histories and likely to be under very different evolutionary pressures. The DNA B genome originated as a satellite that was captured by the monopartite progenitor of all extant bipartite begomoviruses and has subsequently evolved to become an essential genome component. More than 133 begomovirus species having monopartite genomes are known: all originate from the Old World. No monopartite begomoviruses native to the New World have yet been identified.

An edge-coloring is called proper if each edge has a single color, and each two edges of the same color have no vertex in common. A proper edge-coloring does not guarantee the existence of a perfect rainbow matching. For example, consider the graph K2,2 \- the complete bipartite graph on 2+2 vertices. Suppose the edges (x1,y1) and (x2,y2) are colored green, and the edges (x1,y2) and (x2,y1) are colored blue.

Worsaae in 1862 in Om Tvedelingen af Steenalderen, previewed in English even before its publication by The Gentleman's Magazine, concerned about changes in typology during each period, proposed a bipartite division of each age: > Both for Bronze and Stone it was now evident that a few hundred years would > not suffice. In fact, good grounds existed for dividing each of these > periods into two, if not more. He called them earlier or later. The three ages became six periods.

Danish archaeology took the lead in defining the Bronze Age, with little of the controversy surrounding the Stone Age. British archaeologists patterned their own excavations after those of the Danish, which they followed avidly in the media. References to the Bronze Age in British excavation reports began in the 1820s contemporaneously with the new system being promulgated by C.J. Thomsen. Mention of the Early and Late Bronze Age began in the 1860s following the bipartite definitions of Worsaae.

Parablepharismea is a class of free-living marine and brackish anaerobic ciliates that form a major clade of obligate anaerobes within the SAL group (Spirotrichea, Armophorea, and Litostomatea), together with the classes Muranotrichea and Armophorea. Parablepharismea are medium to large, elongated ciliates with navicular outline and holotrichous somatic ciliature composed of dikinetids without postciliodesmata. Their oral ciliature is composed of bipartite paroral membrane and adoral zone of membranelles. They host a thick coat of prokaryotic ectosymbionts and cytoplasmic endosymbionts.

POU proteins are eukaryotic transcription factors containing a bipartite DNA binding domain referred to as the POU domain. The various members of the POU family have a wide variety of functions, all of which are related to the function of the neuroendocrine system and the development of an organism. Some other genes are also regulated, including those for immunoglobulin light and heavy chains (Oct-2), and trophic hormone genes, such as those for prolactin and growth hormone (Pit-1).

Ron Aharoni ( ) (born 1952) is an Israeli mathematician, working in finite and infinite combinatorics. Aharoni is a professor at the Technion – Israel Institute of Technology, where he received his Ph.D. in mathematics in 1979. With Nash-Williams and Shelah he generalized Hall's marriage theorem by obtaining the right transfinite conditions for infinite bipartite graphs. He subsequently proved the appropriate versions of the Kőnig theorem and the Menger theorem for infinite graphs (the latter with Eli Berger).

Like many Hydrozoa, Velella velella has a bipartite life cycle, with a form of alternation of generations. The deep blue, by-the-wind sailors that are recognized by many beach-goers are the polyp phase of the life cycle. Each "individual" with its sail is really a hydroid colony, with many polyps that feed on ocean plankton. These are connected by a canal system that enables the colony to share whatever food is ingested by individual polyps.

In particular, all dual graphs, for all the different planar embeddings of , have isomorphic graphic matroids.. For nonplanar surface embeddings, unlike planar duals, the dual graph is not generally an algebraic dual of the primal graph. And for a non-planar graph , the dual matroid of the graphic matroid of is not itself a graphic matroid. However, it is still a matroid whose circuits correspond to the cuts in , and in this sense can be thought of as a generalized algebraic dual of .. The duality between Eulerian and bipartite planar graphs can be extended to binary matroids (which include the graphic matroids derived from planar graphs): a binary matroid is Eulerian if and only if its dual matroid is bipartite.. The two dual concepts of girth and edge connectivity are unified in matroid theory by matroid girth: the girth of the graphic matroid of a planar graph is the same as the graph's girth, and the girth of the dual matroid (the graphic matroid of the dual graph) is the edge connectivity of the graph..

For t = 2, and for infinitely many values of n, a bipartite graph with n vertices on each side, Ω(n3/2) edges, and no K2,2 may be obtained as the Levi graph of a finite projective plane, a system of n points and lines in which each two points belong to a unique line and each two lines intersect in a unique point. The graph formed from this geometry has a vertex on one side of its bipartition for each point, a vertex on the other side of its bipartition for each line, and an edge for each incidence between a point and a line. The projective planes defined from finite fields of order p lead to K2,2-free graphs with n = p2 + p + 1 and with (p2 + p + 1)(p + 1) edges. For instance, the Levi graph of the Fano plane gives rise to the Heawood graph, a bipartite graph with seven vertices on each side, 21 edges, and no 4-cycles, showing that z(7; 2) ≥ 21\.

The CMS would defray the other half of the price. The Evangelical Jerusalem's Foundation and William II, German Emperor, subscribed each for 180 Napoléons d’or, thus covering the German share. The cession of the reserve land had been negotiated by a bipartite mixed body. So in November 1905 Bussmann proposed to institutionalise the administration and financing of the burialground by a statute of Mount Zion Cemetery, which met consensual agreement, so that the mixed body executed the statute on 25 November 1905.

The new law established under British mandate provided for legal entities as proprietors of real estate. However, until 1936 the purchase of a new cemetery did not progress. The Evangelical congregation of German language feared to lose its equal say and therefore proposed to form two burial boards, one like the then existing bipartite for Mount Zion Cemetery and one quinquipartite for the new Protestant cemetery. The Anglican representatives, however, preferred a single quinquipartite board, now called board of cemetery, for both cemeteries.

Moldova sent a delegation to participate at the 2016 Summer Paralympics in Rio de Janeiro, Brazil, from 7 to 18 September 2016. This was the Eastern European's country sixth appearance in the Summer Paralympic Games since their debut twenty years prior at the 1996 Summer Paralympics. Moldova sent three athletes to these Games, shot put thrower Oxana Spataur, powerlifter Larisa Marinenkova and short-distance swimmer Alexandr Covaliov. Spataur qualified on merit and Covaliov and Marienkova were invited by the Bipartite Commission.

Togo competed at the 2016 Summer Paralympics held in Rio de Janeiro, Brazil from 7 to 18 September 2016. The country's participation in Rio marked its debut appearance in the quadrennial event, although it had competed in the Summer Olympics nine times since the 1972 Games. The delegation consisted of a single lightweight powerlifter, Aliou Bawa, who qualified by being issued with a Bipartite Commission Invitation spot by the International Paralympic Committee. Bawa was the flag bearer for the opening ceremony.

Also, a role can be owned by multiple users. The role mining problem is to find a minimum set of roles, such that for each user, his roles taken together grant access to all specified resources. The set of users together with the set of resources in the system naturally induces a bipartite graph, whose edges are permissions. Each biclique in this graph is a potential role, and the optimum solutions to the role mining problem are precisely the minimum biclique edge covers .

NDUFAF1 is located on the q arm of chromosome 15 in position 15.1. The NDUFAF1 gene produces a 37.8 kDa protein composed of 327 amino acids. NDUFAF1 is associated to complexes of 600 and 700 kDa. Complex I is structured in a bipartite L-shaped configuration, which is made up of a peripheral matrix arm, consisting of flavoproteins and iron-sulfur proteins involved in electron transfer, and a membrane arm, consisting of mtDNA-encoded subunits involved in ubiquinone reduction and proton pumping.

Grammar classifies a language's lexicon into several groups of words. The basic bipartite division possible for virtually every natural language is that of nouns vs. verbs. The classification into such classes is in the tradition of Dionysius Thrax, who distinguished eight categories: noun, verb, adjective, pronoun, preposition, adverb, conjunction and interjection. In Indian grammatical tradition, Pāṇini introduced a similar fundamental classification into a nominal (nāma, suP) and a verbal (ākhyāta, tiN) class, based on the set of suffixes taken by the word.

252 after George Shabat. For example, take p to be the monomial having only one finite critical point and critical value, both zero. Although 1 is not a critical value for p, it is still possible to interpret p as a Belyi function from the Riemann sphere to itself because its critical values all lie in the set {0,1,∞}. The corresponding dessin d'enfant is a star having one central black vertex connected to d white leaves (a complete bipartite graph K1,d).

This raises the question if some of the coefficients are easy to compute. However the computational problem of computing ar for a fixed r ≥ 1 and a given graph G is #P-hard, even for bipartite planar graphs. No approximation algorithms for computing P(G, x) are known for any x except for the three easy points. At the integer points k=3,4,\dots, the corresponding decision problem of deciding if a given graph can be k-colored is NP-hard.

The total area of the proposed state is 6246 km2 and comprises Banarhat, Bhaktinagar, Birpara, Chalsa, Darjeeling, Jaigaon, Kalchini, Kalimpong, Kumargram, Kurseong, Madarihat, Malbazar, Mirik and Nagarkatta. Unlike the 1980s, GJM has maintained that the struggle for Gorkhaland would be through non-violence and non-cooperation. GJM initially resorted to bandhs, hunger strikes and non- payment of utility bills to further their demand. It was quite enough to get the attention of the State Government, who invited them to Kolkata for bipartite talks.

His parents had to find £40,000 to pay for his 18-month stay there. In Spain, he trained with Emilio Sánchez, former world No. 1 doubles player. Murray was born with a bipartite patella, where the kneecap remains as two separate bones instead of fusing together in early childhood, but was not diagnosed until the age of 16. He has been seen holding his knee due to the pain caused by the condition and has pulled out of events because of it.

In discrete mathematics, the Bregman–Minc inequality, or Bregman's theorem, allows one to estimate the permanent of a binary matrix via its row or column sums. The inequality was conjectured in 1963 by Henryk Minc and first proved in 1973 by Lev M. Bregman. Further entropy-based proofs have been given by Alexander Schrijver and Jaikumar Radhakrishnan. The Bregman–Minc inequality is used, for example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph.

The Court's jurisdiction was largely optional, but there were some situations in which they had "compulsory jurisdiction", and states were required to refer cases to them. That came from three sources: the Optional Clause of the League of Nations, general international conventions and "special bipartite international treaties".Hudson (July 1923) p.121 The Optional Clause was a clause attached to the protocol establishing the court and required all signatories to refer certain classes of dispute to the court, with compulsory judgments resulting.

In coding theory, expander codes form a class of error-correcting codes that are constructed from bipartite expander graphs. Along with Justesen codes, expander codes are of particular interest since they have a constant positive rate, a constant positive relative distance, and a constant alphabet size. In fact, the alphabet contains only two elements, so expander codes belong to the class of binary codes. Furthermore, expander codes can be both encoded and decoded in time proportional to the block length of the code.

An (N, M, D, K, e)-disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than (1 − e)M vertices on the right. An extractor is a related type of graph that guarantees an even stronger property; every (N, M, D, K, e)-extractor is also an (N, M, D, K, e)-disperser.

Alexander Viktorovich Karzanov (, born 1947) is a Russian mathematician known for his work in combinatorial optimization. He is the inventor of preflow-push based algorithms for the maximum flow problem, and the co-inventor of the Hopcroft–Karp–Karzanov algorithm for maximum matching in bipartite graphs. He is a chief researcher at the Federal Research Center "Computer Science and Control" (Institute for System Analysis) of the Russian Academy of Sciences. Karzanov was educated at Moscow State University, completing his doctorate there in 1971.

The theory of graphs of bounded clique-width resembles that for graphs of bounded treewidth, but unlike treewidth allows for dense graphs. If a family of graphs has bounded clique-width, then either it has bounded treewidth or every complete bipartite graph is a subgraph of a graph in the family. Treewidth and clique-width are also connected through the theory of line graphs: a family of graphs has bounded treewidth if and only if their line graphs have bounded clique-width.

Other DD- containing proteins, such as ankyrin, MyD88 and pelle, are probably not directly involved in cell death signalling. DD-containing proteins also have links to innate immunity, communicating with Toll-like receptors through bipartite adapter proteins such as MyD88. The DD superfamily is one of the largest and most studied domain superfamilies. It currently comprises four subfamilies, the death domain (DD) subfamily, the death effector domain (DED) subfamily, the caspase recruitment domain (CARD) subfamily and the pyrin domain (PYD) subfamily.

A claw In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K1,3 (that is, a star graph with three edges, three leaves, and one central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern.

The problem can also be stated in terms of zero-one matrices. The connection can be seen if one realizes that each bipartite graph has a biadjacency matrix where the column sums and row sums correspond to (a_1,\ldots,a_n) and (b_1,\ldots,b_n). The problem is then often denoted by 0-1-matrices for given row and column sums. In the classical literature the problem was sometimes stated in the context of contingency tables by contingency tables with given marginals.

The lethal factor protease is produced and secreted by Bacillus anthracis, the agent of anthrax. Together with protective antigen (PA), LF forms a bipartite toxin, Lethal Toxin. The role of PA is to form a translocation channel that delivers LF into the host cell cytosol, where LF play roles in immune response by cleaving and inactivating MAP kinases. LF also directly cleaves NLRP1B proximal to its N-terminus, it is necessary and sufficient for NLRP1B inflammasome formation and CASP1 activation.

They are defined as surjective homomorphisms (i.e., something maps to each vertex) that are also locally bijective, that is, a bijection on the neighbourhood of each vertex. An example is the bipartite double cover, formed from a graph by splitting each vertex v into v0 and v1 and replacing each edge u,v with edges u0,v1 and v0,u1. The function mapping v0 and v1 in the cover to v in the original graph is a homomorphism and a covering map.

The following is the primary sequence of the long form of GPATCH11: Human GPATCH11 protein sequence: The yellow region depicts the G-patch domain, while the blue region depicts the DUF domain. The protein is rich in glutamic acid and is very highly charged. In addition, it is low in amino acids such as valine, threonine, phenylalanine, and proline. It is a soluble protein and has a nuclear export signal and bipartite nuclear import signal implying that it is localized in the nucleus.

DUF2373 is a domain of unknown function found in the C7orf50 protein. This is a highly conserved c-terminal region found from fungi to humans. As for motifs, a bipartite nuclear localization signal (NLS) was predicted from aa6 to aa21, meaning that C7orf50 is likely localized in the nucleus. Interestingly, a nuclear export signal (NES) is also found within the C7orf50 protein at the following amino acids: 150, and 153 - 155, suggesting that C7orf50 has function both inside and outside the nucleus.

A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to represent factorization of a probability distribution function, enabling efficient computations, such as the computation of marginal distributions through the sum-product algorithm. One of the important success stories of factor graphs and the sum-product algorithm is the decoding of capacity- approaching error-correcting codes, such as LDPC and turbo codes. Factor graphs generalize constraint graphs.

The constructive proof described above provides an algorithm for producing a minimum vertex cover given a maximum matching. Thus, the Hopcroft–Karp algorithm for finding maximum matchings in bipartite graphs may also be used to solve the vertex cover problem efficiently in these graphs.For this algorithm, see , p 319, and for the connection to vertex cover see p. 342. Despite the equivalence of the two problems from the point of view of exact solutions, they are not equivalent for approximation algorithms.

There is also an alternative graph-theoretic description of the same lattice: the independent sets of any bipartite graph may be given a partial order in which one independent set is less than another if they differ by removing elements from one side of the bipartition and adding elements to the other side of the bipartition; with this order, the independent sets form a distributive lattice,. and applying this construction to a path graph results in the lattice associated with the Fibonacci cube.

In 1974, Fan Chung graduated from the University of Pennsylvania and became a member of Technical Staff working for the Mathematical Foundations of Computing Department at Bell Laboratories in Murray Hill, New Jersey. She worked under Henry Pollak. During this time, Chung collaborated with many leading mathematicians who work for Bell Laboratories such as Ron Graham. In 1975, Chung published her first joint paper with Graham on Multicolor Ramsey Numbers for Complete Bipartite Graphs which was published in the Journal of Combinatorial Theory.

Despite her initial closeness to the UPG, she distanced herself from this organization. Thus, during the stage of the Galician bipartite (2005–2009), the union maintained a confrontation with the Xunta de Galicia's Ministry of Rural Affairs, in the hands of the UPG. Senra was integrated into the current Encontro Irmandiño in 2009, sponsored by Xosé Manuel Beiras. Senra left the Galego Nationalist Bloc in 2012, at the same time as the supporters of Beiras, who later formed the Anova–Nationalist Brotherhood.

Reformation theologian John Calvin is often quoted as being in support of a bipartite view. Calvin held that while the soul and the spirit are often used interchangeably in the Bible, there are also subtle differences when the two terms are used together. Some have held that the soul and the spirit are interchangeable and the inner life is expressed in a form of literary parallelism. Such parallelism can be found elsewhere in Scripture, such as the Psalms and the Proverbs.

In graph theory, a branch of mathematics, a -biclique-free graph is a graph that has no 2-vertex complete bipartite graph as a subgraph. A family of graphs is biclique-free if there exists a number such that the graphs in the family are all -biclique-free. The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and have also been used in parameterized complexity.

Automatic document classification tasks can be divided into three sorts: supervised document classification where some external mechanism (such as human feedback) provides information on the correct classification for documents, unsupervised document classification (also known as document clustering), where the classification must be done entirely without reference to external information, and semi-supervised document classification, Rossi, R. G., Lopes, A. d. A., and Rezende, S. O. (2016). Optimization and label propagation in bipartite heterogeneous networks to improve transductive classification of texts. Information Processing & Management, 52(2):217–257.

The Meyniel graphs contain the chordal graphs, the parity graphs, and their subclasses the interval graphs, distance-hereditary graphs, bipartite graphs, and line perfect graphs.Meyniel graphs, Information System on Graph Classes and their Inclusions, retrieved 2016-09-25. The house graph is perfect but not Meyniel Although Meyniel graphs form a very general subclass of the perfect graphs, they do not include all perfect graphs. For instance the house graph (a pentagon with only one chord) is perfect but is not a Meyniel graph.

Doctor Light is a bipartite character, comprising supervillain Arthur Light and superhero Jacob Finlay, appearing in comic books published by DC Comics. His stint as Doctor Light is concurrent with that of a superheroine using the same name and nearly identical costume, Kimiyo Hoshi. In 2009, Doctor Light was ranked as IGN's 84th-greatest comic book villain of all time. He made his live-adaptation debut in the series in one episode of Lois & Clark: The New Adventures of Superman, played by David Bowe.

The gene encodes a monomeric protein which shares a 70% amino acid sequence identity, as well as conserved chain folds and flavin adenine dinucleotide (FAD)-binding site structures, with MAO-B. However, MAO-A has a monopartite cavity of approximately 550 angstroms, compared to the 290-angstrom bipartite cavity in MAO-B. Nonetheless, both proteins adopt dimeric forms when membrane-bound. The C-terminal domain of MAO-A forms helical tails which are responsible for attaching the protein to the outer mitochondrial membrane (OMM).

Solving Havannah is PSPACE- complete with respect to the size of the input graph. The proof is by a reduction from generalized geography and is based on using ring-threats to represent the geography graph. In detail, since Lichtenstein and Sipser have proved that generalized geography remained PSPACE-hard even if the graph is only bipartite and of degree at most 3, it only remains to construct an equivalent Havannah position from such a graph, which is accomplished by constructing various gadgets in Havannah.

Almost simultaneously with the ratification of the treaty of Bonn in 1975, two regional committees were founded: The Comité bipartite for the northern part and the Comité tripartite for the southern part. However, in the following years the division proved to be increasingly ineffective. In 1991, the two committees merged and formed the german-french-swiss upper-rhine conference. The legal status and its official role in the cross-border cooperation was finally established in the german-french-swiss treaty of Basel in 2000.

In the mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive (see below).

The site previously contained a medieval church of San Paolo and an Oratory of San Antonio. The two former structures, still evident in the stone around the portals and housing walled up oculi, were hidden behind the tall bipartite facade with giant order pilasters and an unusual tympanum. The tympanum recalls the church of the Babino Gesu all'Esquilino in Rome (now belonging to The Oblate Sisters of the Holy Child Jesus). The Roman church was completed in 1713 with a contribution also by Fuga.

The viruses of this genus have segmented, bipartite genomes that add up to 7,500–19,500 nucleotides in length. Their genomes also code for proteins that do not form part of the virion particles as well as structural proteins. The Universal Virus Database describes that their genome sequences near their 3'-ends are capable of hairpin-loop formation and also believe that their 5'-ends may have methylated caps. Each of the viral RNA molecules contains four hair-pin structures and a pseudoknot in the 3'UTR.

Although the classical Gale–Shapley algorithm cannot be implemented as a comparator circuit, Subramanian came up with a different algorithm showing that the problem is in CC. The problem is also CC-complete. Another problem which is CC-complete is lexicographically-first maximal matching. In this problem, we are given a bipartite graph with an order on the vertices, and an edge. The lexicographically-first maximal matching is obtained by successively matching vertices from the first bipartition to the minimal available vertices from the second bipartition.

It has since been extended to other small values of , and the Zarankiewicz conjecture is known to be true for the complete bipartite graphs with . The conjecture is also known to be true for , , and . If a counterexample exists, that is, a graph requiring fewer crossings than the Zarankiewicz bound, then in the smallest counterexample both and must be odd. For each fixed choice of , the truth of the conjecture for all can be verified by testing only a finite number of choices of .

The qualification slot is allocated to the NPC not to the individual athlete. In case of a Bipartite Commission Invitation the slot is allocated to the individual athlete not to the NPC. To ensure all medal events on the program are viable at the Rio 2016 Paralympic Games IPC Swimming reserves the right to allocate slots to the exclusive use of certain sport classes, in particular for athletes with high support needs. The slot shall be used as allocated or the NPC must return the slot.

There were approximately 30 international conventions under which the Court had similar jurisdiction, including the Treaty of Versailles, the Air Navigation Convention, the Treaty of St. Germain and all mandates signed by the League of Nations.Hudson (January 1923) p.24 It was also foreseen that there would be clauses inserted in bipartite international treaties, which would allow the referral of disputes to the Court; that occurred, with such provisions found in treaties between Czechoslovakia and Austria, and between Czechoslovakia and Poland.Hudson (July 1923) p.

As show, crown graphs are one of a small number of different types of graphs that can occur as distance- regular circulant graphs. describe polygons that have crown graphs as their visibility graphs; they use this example to show that representing visibility graphs as unions of complete bipartite graphs may not always be space- efficient. A crown graph with 2n vertices, with its edges oriented from one side of the bipartition to the other, forms the standard example of a partially ordered set with order dimension n.

Komlós, G. N. Sárközy, E. Szemerédi: Blow-up Lemma, "Combinatorica", 17 (1), 1997, pp. 109-123J. Komlós, G. N. Sárközy, E. Szemerédi: An algorithmic version of the Blow-up Lemma, "Random Structures and Algorithms", 12, 1998, pp. 297-312 in which, together with János Komlós and Endre Szemerédi he proved that the regular pairs in Szemerédi regularity lemma behave like complete bipartite graphs under the correct conditions. The lemma allowed for deeper exploration into the nature of embeddings of large sparse graphs into dense graphs.

Recruitment is the number of individuals in a given species that can survive within a certain amount of time following larval habitation. The higher the level of recruitment, the better chance a larva has of surviving long enough to become an adult fish. Large food supplies, low predator threats, and the availability of nearby anemones are all factors that affect their recruitment levels. A. percula, like most coral reef fish, has a bipartite lifeycle, which has a scattering pelagic larval stage, whereas its resident phase is motionless.

However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings. A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm. The number of perfect matchings in a complete graph Kn (with n even) is given by the double factorial (n − 1)!!.. The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers..

A closely related theorem by states that an n-vertex strongly connected digraph with the property that, for every two nonadjacent vertices u and v, the total number of edges incident to u or v is at least 2n − 1 must be Hamiltonian. Ore's theorem may also be strengthened to give a stronger conclusion than Hamiltonicity as a consequence of the degree condition in the theorem. Specifically, every graph satisfying the conditions of Ore's theorem is either a regular complete bipartite graph or is pancyclic .

In graph theory, a star Sk is the complete bipartite graph K1,k: a tree with one internal node and k leaves (but no internal nodes and k + 1 leaves when k ≤ 1). Alternatively, some authors define Sk to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves. A star with 3 edges is called a claw. The star Sk is edge-graceful when k is even and not when k is odd.

The main periods of 20th century literature are captured in the bipartite division, Modernist literature and Postmodern literature, flowering from roughly 1900 to 1940 and 1945 to 1980 respectively, divided, as a rule of thumb, by World War II. Popular literature develops its own genres such as fantasy and science fiction. Ignored by mainstream literary criticism, these genres develop their own establishments and critical awards, such as the Nebula Award (since 1965), the British Fantasy Award (since 1971) or the Mythopoeic Awards (since 1971).

Well-covered graphs were defined and first studied by . The well-covered graphs include all complete graphs, balanced complete bipartite graphs, and the rook's graphs whose vertices represent squares of a chessboard and edges represent moves of a chess rook. Known characterizations of the well-covered cubic graphs, well-covered claw-free graphs, and well-covered graphs of high girth allow these graphs to be recognized in polynomial time, but testing whether other kinds of graph are well-covered is a coNP-complete problem.

The \lceil r\cdot n\rceil is the smallest possible when n is an integer. An r-partite hypergraph is an r-uniform hypergraph in which the vertices are partitioned into r disjoint sets and each hyperedge contains exactly one vertex of each set (so a 2-partite hypergraph is a just bipartite graph). Let n be any positive integer. Any family of rn-r+1 fractional-matchings (=colors) of size at least n in an r-partite hypergraph has a rainbow-fractional-matching of size n.

1\. Suppose the agents have cardinal utility functions on items. Then, the problem of deciding whether a proportional allocation exists is NP-complete: it can be reduced from the partition problem. 2\. Suppose the agents have ordinal rankings on items, with or without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial time: it can be reduced to the problem of checking whether a bipartite graph admits a feasible b-matching (a matching when the edges have capacities).

Any graph which can be embedded in a plane can also be embedded in a torus. A toroidal graph of genus 1 can be embedded in a torus but not in a plane. The Heawood graph, the complete graph K7 (and hence K5 and K6), the Petersen graph (and hence the complete bipartite graph K3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal.

The experiments were conducted by using a circular split screen (a bipartite field) 2 degrees in diameter, which is the angular size of the human fovea. On one side a test color was projected while on the other an observer-adjustable color was projected. The adjustable color was a mixture of three primary colors, each with fixed chromaticity, but with adjustable brightness. The observer would alter the brightness of each of the three primary beams until a match to the test color was observed.

Highly conserved intracellular domains consisting of a bipartite segment which functions as a GTPase-Activating Protein (GAP). Plexin is the only known receptor molecule to have a GAP domain. In the inactive state, these two sections are separated by a Rho-GTPase binding domain (RBD). When the RBD bind to a Rnd-family Rho-GTPases along with plexin dimerization and semaphoring binding, the intracellular segment undergoes conformational changes which allow the separate GAP domains to interact and become active in turning Rap family Rho-GTPases.

Two planar graphs can have isomorphic medial graphs only if they are dual to each other.. A planar graph with four or more vertices is maximal (no more edges can be added while preserving planarity) if and only if its dual graph is both 3-vertex-connected and 3-regular.. A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. A Hamiltonian cycle in a planar graph corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees. In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees.. If a planar graph has Tutte polynomial , then the Tutte polynomial of its dual graph is obtained by swapping and . For this reason, if some particular value of the Tutte polynomial provides information about certain types of structures in , then swapping the arguments to the Tutte polynomial will give the corresponding information for the dual structures.

Gabriella Campadelli-Fiume took part in a study showing that glycoproteins gD, gB, and gH/gL permit herpes virus to enter cells and were involved in fusion. Their research indicated that gD is not required for the interactions between the glycoproteins gB and gH/gL. Rather, their interactions are made possible due to the multiple sites carried by gB. The multiple sites of gB can be found in the pleckstrin-domain that carries the bipartite fusion loop, and interaction with gH/gL can take place successfully without gD.

A planar graph is an undirected graph that can be embedded into the Euclidean plane without any crossings. A planar graph is called polyhedral if and only if it is 3-vertex-connected, that is, if there do not exist two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite if its vertices can be colored with two different colors such that each edge has one endpoint of each color. A graph is cubic (or 3-regular) if each vertex is the endpoint of exactly three edges.

Literature of the 20th century refers to world literature produced during the 20th century (1901 to 2000). In terms of the Euro-American tradition, the main periods are captured in the bipartite division, Modernist literature and Postmodern literature, flowering from roughly 1900 to 1940 and 1960 to 1990Lewis, Barry. "Postmodernism and Literature." The Routledge Companion to Postmodernism NY: Routledge, 2002, p. 121. respectively, divided, as a rule of thumb, by World War II. The somewhat malleable term of contemporary literature is usually applied with a post-1960 cut off point.

A simple technical way to solve this problem is to extend the input graph to a complete bipartite graph, by adding artificial edges with very large weights. These weights should exceed the weights of all existing matchings, to prevent appearance of artificial edges in the possible solution. As shown by Mulmuley, Vazirani and Vazirani, the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph. Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least ½.

Beta-catenin-like protein 1 is a protein that in humans is encoded by the CTNNBL1 gene. The protein encoded by this gene contains an acidic domain, a putative bipartite nuclear localization signal, a nuclear export signal, a leucine-isoleucine zipper, and phosphorylation motifs. In addition, the encoded protein contains Armadillo/beta-catenin-like repeats, which have been implicated in protein-protein interactions. Although the function of this protein has not been determined, the C-terminal portion of the protein has been shown to possess apoptosis-inducing activity.

Let H1 and H2 be Hilbert spaces of finite dimensions n and m respectively. L(Hi) will denote the space of linear operators acting on Hi. Consider a bipartite quantum system whose state space is the tensor product : H = H_1 \otimes H_2. An (un-normalized) mixed state ρ is a positive linear operator (density matrix) acting on H. A linear map Φ: L(H2) → L(H1) is said to be positive if it preserves the cone of positive elements, i.e. A is positive implied Φ(A) is also.

His Meditationes Sanctorum Patrum, a bipartite collection of prayers purportedly based on writings of Augustine, Bernard of Clairvaux and Anselm of Canterbury (though actually these texts were probably pseudo-Augustinian and -Bernardian, written much later in the style of the Church Fathers), provided Johann Heermann with a basis for many of the hymns in his Devoti musica cordis. Johann Sebastian Bach wrote two chorale cantatas on hymns by Moller or attributed to him, Nimm von uns, Herr, du treuer Gott, BWV 101, and Ach Gott, wie manches Herzeleid, BWV 3.

Every complete graph is a cograph, with a cotree consisting of a single 1-node and leaves. Similarly, every complete bipartite graph is a cograph. Its cotree is rooted at a 1-node which has two 0-node children, one with leaf children and one with leaf children. A Turán graph may be formed by the join of a family of equal-sized independent sets; thus, it also is a cograph, with a cotree rooted at a 1-node that has a child 0-node for each independent set.

In a standard bipartite Bell experiment, Alice's (Bob's) setting x (y), together with her (his) local variable \lambda_A (\lambda_B), influence her (his) local outcome a (b). Bell's theorem can thus be interpreted as a separation between the quantum and classical predictions in a type of causal structures with just one hidden node (\lambda_A,\lambda_B). Similar separations have been established in other types of causal structures. The characterization of the boundaries for classical correlations in such extended Bell scenarios is challenging, but there exist complete practical computational methods to achieve it.

Cryptophyte flagella are inserted parallel to one another, and are covered by bipartite hairs called mastigonemes, formed within the endoplasmic reticulum and transported to the cell surface. Small scales may also be present on the flagella and cell body. The mitochondria have flat cristae, and mitosis is open; sexual reproduction has also been reported. The group have evolved a whole range of light-absorbing pigments, called phycobilins, which are able to absorb wavelengths that's not accessible to other plants or algae, allowing them to live in a variety of different ecological niches.

In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings". A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite.

His church denies the authority of Brigham Young, who led the majority of Latter Day Saints to Utah after Smith's death, together with that of Joseph Smith III, who became the leader of the Reorganized Church of Jesus Christ of Latter Day Saints. The church also disavows plural marriage, and is unique in the Latter Day Saint movement in that it teaches the existence of a bipartite god (God the Father and Jesus Christ), as opposed to the usual three part godhead of Mormonism (God the Father, Jesus Christ, and the Holy Spirit).

In (Monson, Pisanski, Schulte, Ivic-Weiss 2007), the Gray graph appears as a different sort of Levi graph for the edges and triangular faces of a certain locally toroidal abstract regular 4-polytope. It is therefore the first in an infinite family of similarly constructed cubic graphs. As with other Levi graphs, it is a bipartite graph, with the vertices corresponding to points on one side of the bipartition and the vertices corresponding to lines on the other side. Marušič and Pisanski (2000) give several alternative methods of constructing the Gray graph.

It has book thickness 3 and queue number 2.Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018 It can be embedded in the genus-3 orientable surface (which can be represented as the Klein quartic), where it forms the "Klein map" with 24 heptagonal faces, Schläfli symbol {7,3}8. According to the Foster census, the Klein graph, referenced as F056B, is the only cubic symmetric graph on 56 vertices which is not bipartite.. It can be derived from the 28-vertex Coxeter graph.

The concept of incidence coloring was introduced by Brualdi and Massey in 1993 who bounded it in terms of Δ(G). Initially, the incidence chromatic number of trees, complete bipartite graphs and complete graphs was found out. They also conjectured that all graphs can have an incidence coloring using Δ(G) + 2 colors (Incidence coloring conjecture - ICC). This conjecture was disproved by Guiduli, who showed that incidence coloring concept is a directed star arboricity case,Algor I., Alon N. (1989); "The star arboricity of graphs", Discrete Mathematics 75, pp. 11-22.

Because of the correspondence between Eulerian and bipartite matroids among the binary matroids, the binary matroids that are Eulerian may be characterized in alternative ways. The characterization of is one example; two more are that a binary matroid is Eulerian if and only if every element belongs to an odd number of circuits, if and only if the whole matroid has an odd number of partitions into circuits.. collect several additional characterizations of Eulerian binary matroids, from which they derive a polynomial time algorithm for testing whether a binary matroid is Eulerian..

It therefore illustrates the historic process of urban development and, in this particular case, of non- development due to economic circumstances in the 1840s. It demonstrates how early settlers of modest means were able to acquire land and develop it gradually prior to the economic boom of the late 1850s. Its bipartite division demonstrates a closer settlement and development of the land and together with Samson's Cottage wall remains illustrates the development of Sydney at ever higher densities. The loss of the rear yards to public use has destroyed evidence of its former residential use.

Möbius ladders are vertex-transitive – they have symmetries taking any vertex to any other vertex – but (again with the exception of M6) they are not edge-transitive. The edges from the cycle from which the ladder is formed can be distinguished from the rungs of the ladder, because each cycle edge belongs to a single 4-cycle, while each rung belongs to two such cycles. Therefore, there is no symmetry taking a cycle edge to a rung edge or vice versa. When n ≡ 2 (mod 4), Mn is bipartite.

The strongly regular graphs with λ = 0 are triangle free. Apart from the complete graphs on less than 3 vertices and all complete bipartite graphs the seven listed above (pentagon, Petersen, Clebsch, Hoffman-Singleton, Gewirtz, Mesner-M22, and Higman-Sims) are the only known ones. Strongly regular graphs with λ = 0 and μ = 1 are Moore graphs with girth 5. Again the three graphs given above (pentagon, Petersen, and Hoffman-Singleton), with parameters (5, 2, 0, 1), (10, 3, 0, 1) and (50, 7, 0, 1), are the only known ones.

E. Sampathkumar (born 10 June 1936) is a professor emeritus of graph theory from University of Mysore. He has contributed to domination number, bipartite double cover, and reconstruction theory. He was chairman of the department of mathematics of the Karnataka university, DharwarKarnataka University and the University of Mysore (1992–95). Born and raised in Mallur village, Channapatna Taluk, Ramanagaram (District), outside of Bangalore, he earned a M.Sc. (1955) in mathematics from Central College of Bangalore University, and a Ph.D. (1965) in mathematics (Some studies in Boolean algebra) from the Karnataka University, Dharwar.

The smaller of the two Ellingham–Horton graphs Mark Norman Ellingham is a professor of mathematics at Vanderbilt University whose research concerns graph theory.Faculty profile, Vanderbilt U. Mathematics, retrieved 2015-02-10. With Joseph D. Horton, he is the discoverer and namesake of the Ellingham–Horton graphs, two cubic 3-vertex-connected bipartite graphs that have no Hamiltonian cycle.. Ellingham earned his Ph.D. in 1986 from the University of Waterloo under the supervision of Lawrence Bruce Richmond. In 2012, he became one of the inaugural fellows of the American Mathematical Society.

This trait is possibly associated with early tetrapod evolution, which probably also appears on other members of this family and can act as a link to anuran tympanum evolution. The narrow head and elongated snout of Stanocephalosaurus suggests that stress levels during biting are slightly higher than temnospondyls with a wider and shorter skull. Its skull also has an elongated preorbital region compared to other mastodonsaurids. The vertebrae of Stanocephalosaurus are rhachitomous, with a neural arch and a bipartite centrum that is divided into a large, unpaired wedge-shaped intercentrum and smaller paired pleurocentra.

In the mathematical field of graph theory, the Desargues graph is a distance- transitive cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases. The name "Desargues graph" has also been used to refer to a ten-vertex graph, the complement of the Petersen graph, which can also be formed as the bipartite half of the 20-vertex Desargues graph..

Cleavage at the CFCS separates the mature protein from the EHP, allowing it to incorporate into nascent ZP filaments. A variation in the last exon of this gene has previously served as the basis for an additional ZP3 locus; however, sequence and literature review reveals that there is only one full-length ZP3 locus in the human genome. Another locus encoding a bipartite transcript designated POMZP3 contains a duplication of the last four exons of ZP3, including the above described variation, and maps closely to this gene. Orthologs of these genes are found throughout Vertebrata.

Set covering is equivalent to the hitting set problem. That is seen by observing that an instance of set covering can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, the universe represented by vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right- vertices. In the Hitting set problem, the objective is to cover the left- vertices using a minimum subset of the right vertices.

This leads to the speculation that every graph needs either Δ(G) + 1 or Δ(G) + 2 colors, but never more: :Total coloring conjecture (Behzad, Vizing). \chi(G) \le \Delta(G)+2. Apparently, the term "total coloring" and the statement of total coloring conjecture were independently introduced by Behzad and Vizing in numerous occasions between 1964 and 1968 (see Jensen & Toft). The conjecture is known to hold for a few important classes of graphs, such as all bipartite graphs and most planar graphs except those with maximum degree 6.

The Petersen graph is nonplanar. Any nonplanar graph has as minors either the complete graph K_5, or the complete bipartite graph K_{3,3}, but the Petersen graph has both as minors. The K_5 minor can be formed by contracting the edges of a perfect matching, for instance the five short edges in the first picture. The K_{3,3} minor can be formed by deleting one vertex (for instance the central vertex of the 3-symmetric drawing) and contracting an edge incident to each neighbor of the deleted vertex.

However, the same problem is NP- complete when the input may be a mixed graph.. It is #P-complete to count the number of strong orientations of a given graph , even when is planar and bipartite.. However, for dense graphs (more specifically, graphs in which each vertex has a linear number of neighbors), the number of strong orientations may be estimated by a fully polynomial-time randomized approximation scheme.. The problem of counting strong orientations may also be solved exactly, in polynomial time, for graphs of bounded treewidth.

The second part, (as it was in the beginning) repeats material from the beginning of the work but shortened, as a frame. Jones points out that the "wittiness" of it was already used by Monteverdi. Jones remarks that Bach observes a pattern of a bipartite structure of firstly contrasting homophonic blocks and "florid triplet rhythms", secondly "a lighter, quicker conclusion in triple time". He remarks that Bach used a similar pattern again the following year in the for Christmas 1724 which later was included as the of the Mass in B minor.

Yellow mosaic disease of many legumes in India and other South Asian countries is transmitted by geminiviruses belonging to the family Geminiviridae and genus the Begomovirus. Four species, Mungbean yellow mosaic virus (MYMV), Mungbean yellow mosaic India virus (MYMIV), Dolichos yellow mosaic virus (DYMV) and Horsegram yellow mosaic virus (HYMV), are known to cause yellow mosaic disease in different leguminous species. Mungbean yellow mosaic India virus (MYMIV) is a bipartite Begomovirus from the family Geminiviridae and is reported to be the causative agent of the disease in India.

Dorwin Cartwright & Frank Harary (1979) "Balance and clusterability: An overview", pages 25 to 50 in Perspectives in Social Network Research, editors: Paul W. Holland & Samuel Leinhardt, Academic Press The theorem was published by Harary in 1953. It generalizes the theorem that an ordinary (unsigned) graph is bipartite if and only if every cycle has even length. A simple proof uses the method of switching. To prove Harary's theorem, one shows by induction that Σ can be switched to be all positive if and only if it is balanced.

Crown graphs with six, eight, and ten vertices. The outer cycle of each graph forms a Hamiltonian cycle; the eight and ten-vertex graphs also have other Hamiltonian cycles. Solutions to the ménage problem may be interpreted in graph-theoretic terms, as directed Hamiltonian cycles in crown graphs. A crown graph is formed by removing a perfect matching from a complete bipartite graph Kn,n; it has 2n vertices of two colors, and each vertex of one color is connected to all but one of the vertices of the other color.

In addition to his research work, he is well known for his books on algorithms and formal languages coauthored with Jeffrey Ullman and Alfred Aho, regarded as classic texts in the field. In 1986 he received the Turing Award (jointly with Robert Tarjan) "for fundamental achievements in the design and analysis of algorithms and data structures." Along with his work with Tarjan on planar graphs he is also known for the Hopcroft–Karp algorithm for finding matchings in bipartite graphs. In 1994 he was inducted as a Fellow of the Association for Computing Machinery.

As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. In particular, there is no transitive closure of set membership for such hypergraphs. Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer bipartite, but is rather just some general directed graph. The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges.

The complete graph K5 has a RAC drawing with straight edges, but K6 does not. Every 6-vertex RAC drawing has at most 14 edges, but K6 has 15 edges, too many to have a RAC drawing. A complete bipartite graph Ka,b has a RAC drawing with straight edges if and only if either min(a,b) ≤ 2 or a + b ≤ 7\. If min(a,b) ≤ 2, then the graph is a planar graph, and (by Fáry's theorem) every planar graph has a straight-line drawing with no crossings.

The problem of developing an online algorithm for matching was first considered by Richard M. Karp, Umesh Vazirani, and Vijay Vazirani in 1990. In the online setting, nodes on one side of the bipartite graph arrive one at a time and must either be immediately matched to the other side of the graph or discarded. This is a natural generalization of the secretary problem and has applications to online ad auctions. The best online algorithm, for the unweighted maximization case with a random arrival model, attains a competitive ratio of 0.696.

The laws consist of six predictive statements about the direction of analogical changes: # A bipartite marker tends to replace an isofunctional simple marker. # The directionality of analogy is from a “basic” form to a “subordinate” form with respect to their spheres of usage. # A structure consisting of a basic and a subordinate member serves as a foundation for a basic member which is isofunctional but isolated. # When the old (non-analogical) form and the new (analogical) form are both in use, the former remains in secondary function and the latter takes the basic function.

CiLV-N has short, rod-shaped particles, 120 to 130 nanometers (nm) long and 35 to 40 nm wide, occurring in the nucleus or cytoplasm of the infected cells, and associated with the presence of viroplasm in the nucleus. The CiLV-N genome is a bipartite, negative-sense, single stranded RNA ((-)ssRNA). Both RNAs have 3'-terminal poly(A) tails. CiLV-N RNA1 (6,268 nucleotides (nt)) contains five Open Reading Frames (ORF) encoding the nucleocapsid protein (N), putative phosphoprotein (P), cell-to-cell movement protein (MP), matrix protein (M), and glycoprotein (G).

The H motif is located in the hinge and the ACA motif is located in the tail region; 3 nucleotides from the 3' end of the sequence. The hairpin regions contain internal bulges known as recognition loops in which the antisense guide sequences (bases complementary to the target sequence) are located. These guide sequences essentially mark the location of the uridine on the target rRNA that are going to be modified. This recognition sequence is bipartite (constructed from the two different arms of the loop region) and forms complex pseudo-knots with the target RNA.

The Bip Buffer (bipartite buffer) is very similar to a circular buffer, except it always returns contiguous blocks which can be variable length. This offers nearly all the efficiency advantages of a circular buffer while maintaining the ability for the buffer to be used in APIs that only accept contiguous blocks.Simon Cooke (2003), "The Bip Buffer - The Circular Buffer with a Twist" Fixed-sized compressed circular buffers use an alternative indexing strategy based on elementary number theory to maintain a fixed-sized compressed representation of the entire data sequence.

J.L. Xiang Li and Balázs Szegedy (2011) introduced the idea of using entropy to prove some cases of Sidorenko's conjecture. Szegedy (2015) later applied the ideas further to prove that an even wider class of bipartite graphs have Sidorenko's property. While Szegedy's proof wound up being abstract and technical, Tim Gowers and Jason Long reduced the argument to a simpler one for specific cases such as paths of length 3. In essence, the proof chooses a nice probability distribution of choosing the vertices in the path and applies Jensen's inequality (i.e.

A subdivision of K3,3 in the generalized Petersen graph G(9,2), showing that the graph is nonplanar. In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or of K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).

The use of multipartite entangled states instead of a bipartite maximally entangled state allows for several new features: either the sender can teleport information to several receivers either sending the same state to all of them (which allows to reduce the amount of entanglement needed for the process) or teleporting multipartite states or sending a single state in such a way that the receiving parties need to cooperate to extract the information. A different way of viewing the latter setting is that some of the parties can control whether the others can teleport.

On 31 January 2019, Diamandis teased the new album by posting a picture on her Instagram with the caption "8 Days". The day after, she revealed in an interview that the new album would come out sometime in early 2019. On 6 February 2019, it was revealed that the title of the lead single of the album would be "Handmade Heaven", which was subsequently released two days later. Her bipartite fourth studio album Love + Fear was then released in two halves on 4 April and 26 April 2019.

The Petersen family. As showed, each of the seven graphs of the Petersen family is intrinsically linked: no matter how each of these graphs is embedded in space, they have two cycles that are linked to each other. These graphs include the complete graph K6, the Petersen graph, the graph formed by removing an edge from the complete bipartite graph K4,4, and the complete tripartite graph K3,3,1. Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space.

That is, a graph is a line graph if and only if no subset of its vertices induces one of these nine graphs. In the example above, the four topmost vertices induce a claw (that is, a complete bipartite graph K1,3), shown on the top left of the illustration of forbidden subgraphs. Therefore, by Beineke's characterization, this example cannot be a line graph. For graphs with minimum degree at least 5, only the six subgraphs in the left and right columns of the figure are needed in the characterization.

The two strands are then split apart, and hairpin loops are formed at both ends of both strands. In place of the HUH endonuclease, bidnaviruses encode their own protein-primed DNA polymerase that replicates the genome, which is bipartite and packaged into two separate virions, instead of using the host cell's DNA polymerase for replication. Additionally, some viruses in the realm are dsDNA viruses with circular genomes, including Polyomaviridae and Papillomaviridae, also assigned to the phylum Cossaviricota. Instead of replicating via RCR, these viruses use bidirectional DNA replication.

The vertebrae of the Stanocephalosaurus are rhachitomous, with a neural arch and a bipartite centrum that is divided into a large, unpaired wedge-shaped intercentrum and smaller paired pleurocentra. In anterior and posterior views, the intercentrum is a dorsally half-ring, surrounding the persistent notochord from ventral and lateral sides. Lateral and ventral surfaces of the intercentrum are smooth, suggesting a continuation of cartilage due to the unfinished medial surface. The posterodorsal margin of the intercentrum also shows a parapophysis for articulation with the capitulum of the ribs.

The team that travelled to Italy consisted of Takudzwa Gwariro, Margret Bangajena, Previous Wiri, Chipo Zhento and Davis' daughter Jessica as their able-bodied cox. Their chances were almost dashed at the first hurdle after Zhento suffered a seizure. The team was asked to pull- out, but a Cameroon doctor, who knew the shame that would be brought onto Zheno's family if she caused the team to fail, cleared Zhento to compete. Although the team finished second to last, ahead of Japan, they qualified for Rio after being offered a bipartite invitation.

Single stranded closed circular DNA. Many begomoviruses have a bipartite genome: this means that the genome is segmented into two segments (referred to as DNA A and DNA B) that are packaged into separate particles. Both segments are generally required for successful symptomatic infection in a host cell but DNA B is dependent for its replication upon DNA A, which can in some begomoviruses apparently cause normal infections on its own. Begomovirus The DNA A segment typically encodes five to six proteins including replication protein Rep, coat protein and transport and/or regulatory proteins.

Drisko studied this question using the terminology of Latin rectangles. He proved that, for any n≤k, in the complete bipartite graph Kn,k, any family of 2n-1 matchings (=colors) of size n has a perfect rainbow matching (of size n). He applied this theorem to questions about group actions and difference sets. Drisko also showed that 2n-1 matchings may be necessary: consider a family of 2n-2 matchings, of which n-1 are {(x1,y1), (x2,y2), ..., (xn,yn)} and the other n-1 are {(x1,y2), (x2,y3), ..., (xn,y1)}.

Yana is often classified in the Hokan superstock. Sapir suggested a grouping of Yana within a Northern Hokan sub-family with Karuk, Chimariko, Shastan, Palaihnihan, and Pomoan. Contemporary linguists generally consider Yana to be a language isolate.Marianne Mithun, The Languages of Native North America (1999, Cambridge)Lyle Campbell, American Indian Languages, The Historical Linguistics of Native America (1997, Oxford) The use of bipartite verb stem formation in Yana is not a Hokan characteristic, but is used in other non-Hokan languages in the area, suggesting that Yana has stayed geographically stable.

The intersection of two or more matroids is the family of sets that are simultaneously independent in each of the matroids. The problem of finding the largest set, or the maximum weighted set, in the intersection of two matroids can be found in polynomial time, and provides a solution to many other important combinatorial optimization problems. For instance, maximum matching in bipartite graphs can be expressed as a problem of intersecting two partition matroids. However, finding the largest set in an intersection of three or more matroids is NP-complete.

In 1997, Sós was awarded the Széchenyi Prize. One of her results is the Kővári–Sós–Turán theorem concerning the maximum possible number of edges in a bipartite graph that does not contain certain complete subgraphs. Another is the following so- called friendship theorem proved with Paul Erdős and Alfréd Rényi: if, in a finite graph, any two vertices have exactly one common neighbor, then some vertex is joined to all others. In number theory, Sós proved the three-gap theorem, conjectured by Hugo Steinhaus and proved independently by Stanisław Świerczkowski.

The worst-case complexity of DSatur is Ο(n2), however in practical some additional expenses result from the need for holding the degree of saturation of the uncoloured vertices. DSatur has been proven to be exact for bipartite graphs, as well as for cycle and wheel graphs. In an empirical comparison by Lewis 2015, DSatur produced significantly better vertex colourings than the greedy algorithm on random graphs with edge probability p = 0.5 at varying number of vertices, while in turn producing significantly worse colourings than the Recursive Largest First algorithm.

The first multipartite entanglement measure that is neither a direct generalization nor an easy combination of bipartite measures was introduced by Coffman et al. and called tangle. Definition [tangle]: :\; \tau(A : B : C) = \tau(A : BC) - \tau(AB) - \tau(AC) , where the \; 2-tangles on the right-hand-side are the squares of concurrence. The tangle measure is permutationally invariant; it vanishes on all states that are separable under any cut; it is nonzero, for example, on the GHZ-state; it can be thought to be zero for states that are 3-entangled (i.e.

Bean calico mosaic virus (BCaMV or BCMoV) is a plant virus transmitted by whiteflies that infects bean genera and species within the families Fabaceae, Malvaceae, and Solanaceae. Like other New World begomoviruses, its genome is bipartite, or having two parts. Phylogenetic analysis of its two genome segments, DNA-A and DNA-B, indicate the virus is from Sonora, Mexico and shares a most recent common ancestor with the Leaf curl virus-E strain and the Texas pepper virus, both also found in the Sonora desert, and the Cabbage leaf curl virus from Florida.

The exception of Whitney's theorem: these two graphs are not isomorphic but have isomorphic line graphs. The Whitney graph isomorphism theorem, shown by Hassler Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K3, the complete graph on three vertices, and the complete bipartite graph K1,3, which are not isomorphic but both have K3 as their line graph. The Whitney graph theorem can be extended to hypergraphs.Dirk L. Vertigan, Geoffrey P. Whittle: A 2-Isomorphism Theorem for Hypergraphs.

The DisGeNET Cytoscape plugin offers a network representation of the gene-disease associations. It represents gene-disease associations in terms of bipartite graphs and additionally provides gene centric and disease centric views of the data. It assists the user in the interpretation and exploration of human complex diseases with respect to their genetic origin by a variety of built-in functions. Using the DisGeNET Cytoscape plugin you can perform queries restricted to (i) the original data source, (ii) the association type, (iii) the disorder class of interest and (iv) specific diseases or genes.

In matroid theory, a mathematical discipline, the girth of a matroid is the size of its smallest circuit or dependent set. The cogirth of a matroid is the girth of its dual matroid. Matroid girth generalizes the notion of the shortest cycle in a graph, the edge connectivity of a graph, Hall sets in bipartite graphs, even sets in families of sets, and general position of point sets. It is hard to compute, but fixed-parameter tractable for linear matroids when parameterized both by the matroid rank and the field size of a linear representation.

The well- colored graphs include the complete graphs and odd-length cycle graphs (the graphs that form the exceptional cases to Brooks' theorem) as well as the complete bipartite graphs and complete multipartite graphs. The simplest example of a graph that is not well-colored is a four-vertex path. Coloring the vertices in path order uses two colors, the optimum for this graph. However, coloring the ends of the path first (using the same color for each end) causes the greedy coloring algorithm to use three colors for this graph.

On 25 June 1947, the 52 delegates elected in an indirect ballot of one delegate per 750,000 citizens by the Landtage (parliaments) of the eight Länder in the Bizone gathered in Frankfurt am Main; on 9 August, the law for the reorganisation of the bizonal economic agencies was passed. Soon afterwards, however, the constructional flaws of the Economic Council as a whole came to the fore requiring the reorganisation of the bizonal administration. On 9 February 1948, the Frankfurter Statut defining the changes to the Economic Council came into effect. These were the renaming of the Executive Committee as Länderrat, the creation of a Verwaltungsrat (Administrative Council) formed by the Executive Directors and supervised by a chairman officially titled Oberdirektor, and, finally, the doubling of the delegates in the Economic Council, something which did not affect the proportion of political parties in this second economic parliament. While the Economic Council was a decisive platform for the political debate and factual implementation of any emerging economic concept, the parliament’s resolutions and acts remained subject to the authorisation by the Allied Zwei-Zonen-Amt (Bipartite Board) in Berlin and were controlled by the so-called ‘Zweizonenkontrollamt’ (Bipartite Control Office) (BICO) in Frankfurt.

An order-k folded cube graph is k-regular with 2k − 1 vertices and 2k − 2k edges. The chromatic number of the order-k folded cube graph is two when k is even (that is, in this case, the graph is bipartite) and four when k is odd. provides a proof, and credits the result to Naserasr and Tardif. The odd girth of a folded cube of odd order is k, so for odd k greater than three the folded cube graphs provide a class of triangle-free graphs with chromatic number four and arbitrarily large odd girth.

At the age of 39, Larisa Marinenkova was the oldest athlete to represent Moldova at the Rio Paralympics. She was competing in the Paralympic Games for the third time, having represented Moldova at the 2008 Summer Olympics and the 2012 Summer Paralympics. Marinenkvoa has been affected by cerebral palsy since childhood, resulting in permanent muscle weakness, and works as an engineer at a state-owned enterprise. She attained qualification to the Games by the Paralympic Committee of Moldova being granted an invitation for Marinenkova by the Bipartite Commission to allow her to compete after she did not meet the qualifying standards for powerlifting.

Alternatively, describing the problem using graph theory: :The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is a minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment. If the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called linear assignment.

1700–1100 BC), a collection of early Vedic hymns. Both texts are considered to have a common archaic Indo-Iranian origin. The Gathas portray an ancient Stone-Bronze Age bipartite society of warrior-herdsmen and priests (compared to Bronze tripartite society; some conjecture that it depicts the Yaz culture), and thus it is implausible that the Gathas and Rigveda could have been composed more than a few centuries apart. These scholars suggest that Zoroaster lived in an isolated tribe or composed the Gathas before the 1200–1000 BC migration by the Iranians from the steppe to the Iranian Plateau.

272x272px The protein contains one domain of unknown function, DUF 4709, spanning from the 7th amino acid to the 280th amino acid. Motifs that are predicted to exist include an N-terminal motif, RxxL motif, and KEN conserving motif, which all signal for protein degradation. Another motif that is predicted to exist is a Wxxx motif, which facilitates entrance of PTS1 cargo proteins into the organellar lumen, and a RVxPx motif which allows protein transport from the trans-Golgi network to the plasma membrane of the cilia. There is also a bipartite nuclear localization signal at the end of the protein sequence.

Mithun compiled a comprehensive overview of Native American languages in The Languages of Native North America. A review on the Linguist List describes the work as "an excellent book to have as a reference" and as containing "an incredible amount of information and illustrative data." The work is a bipartite reference organized firstly by grammatical categories (including categories that are particularly widespread in North America, such as polysynthesis), and secondly by family. In 2002 the volume won the Leonard Bloomfield Book Award, awarded annually by the Linguistic Society of America for the best book in linguistics.

A linear-time algorithm for finding a longest path in a tree was proposed by Dijkstra in 1960's, while a formal proof of this algorithm was published in 2002.. Furthermore, a longest path can be computed in polynomial time on weighted trees, on block graphs, on cacti,. on bipartite permutation graphs,. and on Ptolemaic graphs.. For the class of interval graphs, an O(n^4)-time algorithm is known, which uses a dynamic programming approach.. This dynamic programming approach has been exploited to obtain polynomial-time algorithms on the greater classes of circular-arc graphs. and of co-comparability graphs (i.e.

This gene encodes a bipartite protein with distinct amino- and carboxy-terminal domains. The amino-terminus contains nuclear localization signals and the carboxy-terminus contains numerous consecutive sequences with extensive similarity to proteins in the gelsolin family of actin-binding proteins, which cap, nucleate, and/or sever actin filaments. The gene product is tightly associated with both actin filaments and plasma membranes, suggesting a role as a high-affinity link between the actin cytoskeleton and the membrane. Its function may include recruitment of actin and other cytoskeletal proteins into specialized structures at the plasma membrane and in the nuclei of growing cells.

They reject the traditional doctrine of hell as a state of everlasting conscious torment, believing instead that the wicked will be permanently destroyed after the millennium. The theological term for this teaching is Annihilationism. The Adventist views about death and hell reflect an underlying belief in: (a) conditional immortality (or conditionalism), as opposed to the immortality of the soul; and (b) the holistic (or monistic) Christian anthropology or nature of human beings, as opposed to bipartite or tripartite views. Adventist education hence strives to be holistic in nature, involving not just the mind but all aspects of a person.

Each exposed tower elevation has single window with hood to first stage; bipartite window with hood to second stage; blind oculus to third stage; and tripartite with hood and painted timber louvers to (belfry) fourth stage, which is above cornice level and has slightly setback corners. The seven-bay four-story orange brick schoolhouse with stone trim was built 1914. Main façade is detailed with central three-bay breakfront that to ground floor is accentuated in four-centre-arched stone entrances. Ground-floor is double- height and defined from rest of structure with separating stone stringcourse and second floor sillcourse.

For the restricted case of Steiner Tree problem with distances 1 and 2, a 1.25-approximation algorithm is known. Karpinski and Alexander Zelikovsky constructed PTAS for the dense instances of Steiner Tree problems. In a special case of the graph problem, the Steiner tree problem for quasi-bipartite graphs, S is required to include at least one endpoint of every edge in G. The Steiner tree problem has also been investigated in higher dimensions and on various surfaces. Algorithms to find the Steiner minimal tree have been found on the sphere, torus, projective plane, wide and narrow cones, and others.

That is, if Hill's conjecture is correct, then the drawing of this graph that minimizes the number of crossings is a two-page drawing.. The book thickness of the complete bipartite graph is at most . To construct a drawing with this book thickness, for each vertex on the smaller side of the bipartition, one can place the edges incident with that vertex on their own page. This bound is not always tight; for instance, has book thickness three, not four. However, when the two sides of the graph are very unbalanced, with , the book thickness of is exactly .

Social dialogue (or social concertation) is the process whereby social partners (trade unions and employer organisations) negotiate, often in collaboration with the government, to influence the arrangement and development of work-related issues, labour market policies, social protection, taxation or other economic policies. It is a widespread procedure to develop public policies in Western Europe in particular. These can be direct relations between the social partners themselves ("bipartite") or relations between governmental authorities and the social partners ("tripartite"). To make it more clear, Social dialogue can mean negotiation, consultation or simply an exchange of views between representatives of employers, workers and governments.

Eohupehsuchus has the shortest known neck among hupehsuchians, with a cervical series comprising only six vertebrae. The shortness of its neck is the basis of its specific epithet, brevicollis, and distinguishes it from other hupehsuchians, along with the features of the skull roof discussed above. Like other hupehsuchians, Eohupehsuchus has vertebrae with bipartite neural spines consisting of a first segment that begins above the base of the neural arch and a second segment that begins above the first. In Eohupehsuchus, the second segments appear throughout the dorsal series beginning with the third dorsal vertebra, and are absent in the sacral series.

Quantum mechanics may be modelled on a projective Hilbert space, and the categorical product of two such spaces is the Segre embedding. In the bipartite case, a quantum state is separable if and only if it lies in the image of the Segre embedding. Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement""Geometrical aspects of entanglement", Physical Review A 74, 012313 (2006) describe the problem and study the geometry of the separable states as a subset of the general state matrices. This subset have some intersection with the subset of states holding Peres-Horodecki criterion.

Maturation of the notch receptor involves cleavage at the prospective extracellular side during intracellular trafficking in the Golgi complex. This results in a bipartite protein, composed of a large extracellular domain linked to the smaller transmembrane and intracellular domain. Binding of ligand promotes two proteolytic processing events; as a result of proteolysis, the intracellular domain is liberated and can enter the nucleus to engage other DNA-binding proteins and regulate gene expression. Notch and most of its ligands are transmembrane proteins, so the cells expressing the ligands typically must be adjacent to the notch expressing cell for signaling to occur.

Most coastal fish species have a bipartite life cycle where larvae are pelagic before settling out of the plankton to live on a reef. While these fish travel varying distances during their life history, their larvae have the potential to move tens to hundreds of km, more than the more sedentary adults and juveniles, which have home ranges of <1 m to a few km. Adults and juveniles of some species travel tens to hundreds of kilometers as they mature to reach appropriate habitats (e.g., such as coral reef, mangrove and seagrass habitats) or to migrate to spawning areas.

One consequence of Maekawa's theorem is that the total number of folds at each vertex must be an even number. This implies (via a form of planar graph duality between Eulerian graphs and bipartite graphs) that, for any flat-foldable crease pattern, it is always possible to color the regions between the creases with two colors, such that each crease separates regions of differing colors.. See in particular Theorem 3.1 and Corollary 3.2. The same result can also be seen by considering which side of the sheet of paper is uppermost in each region of the folded shape.

All Begomovirus species causing cotton leaf curl disease have geminate particles, approximately 18-20 nm in diameter and 30 nm long and a circular, single- stranded DNA genome. All except Cotton leaf crumple virus have a monopartite genome, with all viral products required for replication, systemic movement and whitefly transmission encoded on a single DNA component of c. 2.75 kB (DNA A). The genome of CLCrV is bipartite. Two smaller, circular, single-stranded DNA molecules, named DNA 1 and DNA β, are associated with a range of monopartite begomoviruses from the Old World including the cotton leaf curl viruses.

Wilde and Brun have integrated the theory of entanglement- assisted stabilizer codes and quantum convolutional codes in a series of articles (Wilde and Brun 2007a, 2007b, 2008, 2009) to form a theory of entanglement-assisted quantum convolutional coding. This theory supposes that a sender and receiver share noiseless bipartite entanglement that they can exploit for protecting a stream of quantum information. (Wilde 2009), building on work of (Ollivier and Tillich 2004) and (Grassl and Roetteler 2006), also showed how to encode these codes with quantum shift register circuits, a natural extension of the theory of classical shift register circuits.

There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure. Another way to define a distance again uses a graph-theoretic notion in a related structure, this time the collinearity graph of the incidence structure.

PCIJ on the ICJ website With the heightened international tension in the 1930s, the Court became less used. By a resolution from the League of Nations on 18 April 1946, both the Court and the League ceased to exist and were replaced by the International Court of Justice and the United Nations. The Court's mandatory jurisdiction came from three sources: the Optional Clause of the League of Nations, general international conventions and special bipartite international treaties. Cases could also be submitted directly by states, but they were not bound to submit material unless it fell into those three categories.

Algorithms for finding strongly connected components may be used to solve 2-satisfiability problems (systems of Boolean variables with constraints on the values of pairs of variables): as showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both contained in the same strongly connected component of the implication graph of the instance.. Strongly connected components are also used to compute the Dulmage–Mendelsohn decomposition, a classification of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching in the graph..

Agnoprotein is typically quite short: examples from BK virus, JC virus, and SV40 are 62, 71, and 66 amino acid residues long, respectively. Among other known polyomavirus genomes with a predicted agnogene, the length of the resulting predicted protein varies considerably, from as short as 30 to as long as 154 residues. It contains a highly hydrophobic central amino acid sequence, a "bipartite" nuclear localization sequence at the N-terminus, and highly basic amino acids at both termini. Comparison of the sequences of different viral agnoproteins suggests sequence conservation toward the N-terminus with greater variability toward the C-terminus.

The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem: :A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph K5 or the complete bipartite graph K3,3 (utility graph). A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •----• to •--•--•) zero or more times. An example of a graph with no K5 or K3,3 subgraph. However, it contains a subdivision of K3,3 and is therefore non-planar.

This second subclass contains an addition subunit (PyrK) containing an iron-sulfur cluster and a flavin adenine dinucleotide (FAD). Meanwhile, Class 2 DHODHs use coenzyme Q/ubiquinones for their oxidant. In higher eukaryotes, this class of DHODH contains an N-terminal bipartite signal comprising a cationic, amphipathic mitochondrial targeting sequence of about 30 residues and a hydrophobic transmembrane sequence. The targeting sequence is responsible for this protein’s localization to the IMM, possibly from recruiting the import apparatus and mediating ΔΨ-driven transport across the inner and outer mitochondrial membranes, while the transmembrane sequence is essential for its insertion into the IMM.

It portrays the female wolf with the two infant brothers Romulus and Remus, illustrating the myth of the founding of Rome, superimposed on a bipartite golden yellow over maroon red shield. In the myth from which the club takes their nickname and logo, the twins (sons of Mars and Rhea Silvia) are thrown into the river Tiber by their uncle Amulius. A she- wolf then saved the twins and looked after them. Eventually, the two twins took revenge on Amulius before falling-out themselves – Romulus killed Remus and was thus made king of a new city named in his honour, Rome.

A Gold color metal and enamel device 1 3/16 inches (3.02 cm) in height overall consisting of a shield per fess wavy, the chief per pale Argent and Sable, the base Gules, a demi-missile in pale of the first (Silver Gray) issuing from a demi-annulet of the first in base, in fess three wavy barrulets Azure; in dexter chief a lightning bolt bendwise of the second and in sinister chief a lightning bolt bend sinisterwise of the first. Around the bottom of the shield is a Black bipartite scroll inscribed with "QUISQUAM" "USQUAM" in Gold letters.

A complete bipartite graph is a circulant graph if it has the same number of vertices on both sides of its bipartition. If two numbers and are relatively prime, then the rook's graph (a graph that has a vertex for each square of an chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group Cmn \simeq Cm×Cn. More generally, in this case, the tensor product of graphs between any - and -vertex circulants is itself a circulant.

For example, if the new variables c, corresponding to the old constraint C(x,y) can assume values (1,2) and (2,0), two new constraints are added: the first one enforces x to take value 1 if c=(1,2) value 2 if c=(2,0), and vice versa. The second condition enforces a similar condition for variable y. The graph representing the result of this transformation is bipartite, as all constraints are between a new and an old variable. Moreover, the constraints are functional: for any given value of a new variable, only one value of the old variable may satisfy the constraint.

In the mathematical field of graph theory, the Ellingham–Horton graphs are two 3-regular graphs on 54 and 78 vertices: the Ellingham–Horton 54-graph and the Ellingham–Horton 78-graph. They are named after Joseph D. Horton and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.. The book thickness of the Ellingham-Horton 54-graph and the Ellingham-Horton 78-graph is 3 and the queue numbers 2Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, Universität Tübingen, 2018.

There is a O(V^{2}E) time algorithm to find a maximum matching or a maximum weight matching in a graph that is not bipartite; it is due to Jack Edmonds, is called the paths, trees, and flowers method or simply Edmonds' algorithm, and uses bidirected edges. A generalization of the same technique can also be used to find maximum independent sets in claw-free graphs. More elaborate algorithms exist and are reviewed by Duan and Pettie (see Table III). Their work proposes an approximation algorithm for the maximum weight matching problem, which runs in linear time for any fixed error bound.

CiLV-N RNA2 (5,847 nt) contains one ORF encoding the RNA-dependent RNA polymerase (RdRp) replication module. The size and structure of the CiLV-N genome closely resembles the genome organization of Orchid fleck virus (OFV) and is likely to be a member of the newly proposed genus Dichorhavirus. CiLV-C has short, membrane-bound bacilliform particles, 120 to 130 nm long and 50 to 55 wide; it is found in the endoplasmic reticulum in the cytoplasm of infected cells, and large electron dense viroplasm is observed in the cytoplasm. CiLV-C has a bipartite, positive-sense, single stranded, RNA ((+)ssRNA) genome.

Bitext word alignment finds out corresponding words in two texts. Bitext word alignment or simply word alignment is the natural language processing task of identifying translation relationships among the words (or more rarely multiword units) in a bitext, resulting in a bipartite graph between the two sides of the bitext, with an arc between two words if and only if they are translations of one another. Word alignment is typically done after sentence alignment has already identified pairs of sentences that are translations of one another. Bitext word alignment is an important supporting task for most methods of statistical machine translation.

Although Kircher's work was not mathematically based, he did develop various systems for generating and counting all combinations of a finite collection of objects (i.e., a finite set), based on the previous work of Ramon Llull. His methods and diagrams are discussed in Ars Magna Sciendi, sive Combinatoria, 1669. They include what may be the first recorded drawings of complete bipartite graphs, extending a similar technique used by Llull to visualize complete graphs.. Kircher also employed combinatorics in his Arca Musarithmica, an aleatoric music composition device capable of producing millions of church hymns by combining randomly selected musical phrases.

Fermitin family homolog 3) (FERMT3), also known as kindlin-3 (KIND3), MIG2-like protein (MIG2B), or unc-112-related protein 2 (URP2) is a protein that in humans is encoded by the FERMT3 gene. The kindlin family of proteins, member of the B4.1 superfamily, comprises three conserved protein homologues, kindlin 1, 2, and 3. They each contain a bipartite FERM domain comprising four subdomains F0, F1, F2, and F3 that show homology with the FERM head (H) domain of the cytoskeletal Talin protein. Kindlins have been linked to Kindler syndrome, leukocyte adhesion deficiency, cancer and other acquired human diseases.

In the media and popular science, quantum non- locality is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations. A well-known example is the Werner states that are entangled for certain values of p_{sym}, but can always be described using local hidden variables. Moreover, it was shown that, for arbitrary numbers of parties, there exist states that are genuinely entangled but admit a local model.

However, genera within phytosaurs may also have had different ecological preferences. Such is the case for Nicrosaurus and Mystriosuchus, the biggest distinguishing factor between the two being the shape of their snouts. The latter had a slender skull with bipartite dentition, suggesting a diet of fish and small tetrapods, while the former had a massive skull with tripartite dentition, suggesting prey were larger animals. Nicrosaurus and Mystriosuchus have both been recovered in the first and second Stubensandstein in arkosic sandstones separated by floodplain mudstones and were both buried during flooding events in a freshwater river habitat.

The qualification period lasts from 1 January to 31 December 2018 and is based on world rankings. In order for cyclists to maintain their slots, they must compete in the 2019 or 2020 UCI Para-cycling Track World Championships or UCI Para-cycling Road World Championships or any UCI Para-Cycling Road World Cup in 2019. The highest ranked athlete per NPC will obtain one qualification spot. The qualification slot is allocated to the NPC not to the individual athlete: in the case of bipartite commission invitations, the slot is allocated to the athlete not to the NPC.

The Wagner graph, an eight-vertex Möbius ladder arising in Wagner's characterization of K5-free graphs. Wagner is known for his contributions to graph theory and particularly the theory of graph minors, graphs that can be formed from a larger graph by contracting and removing edges. Wagner's theorem characterizes the planar graphs as exactly those graphs that do not have as a minor either a complete graph K5 on five vertices or a complete bipartite graph K3,3 with three vertices on each side of its bipartition. That is, these two graphs are the only minor-minimal non-planar graphs.

Naval Enigma used an Indicator to define a key mechanism, with the key being transmitted along with the ciphertext. The starting position for the rotors was transmitted just before the ciphertext, usually after having been enciphered by Naval Enigma. The exact method used was termed the indicator procedure. A properly self-reciprocal bipartite digraphic encryption algorithm was used for the super-encipherment of the indicators (German:Spruchschlüssel) with basic wheel settings The Enigma Cipher Keys called Heimische Gewässer (English Codename: Dolphin), (Plaice), Triton (Shark), Niobe (Narwhal) and Sucker all used the Kenngruppenbuch and bigram tables to build up the Indicator.

An edge coloring of G is a vertex coloring of its line graph L(G), and vice versa. Thus, :\chi'(G)=\chi(L(G)). There is a strong relationship between edge colorability and the graph’s maximum degree \Delta(G). Since all edges incident to the same vertex need their own color, we have : \chi'(G) \ge \Delta(G). Moreover, : Kőnig’s theorem: \chi'(G) = \Delta(G) if G is bipartite. In general, the relationship is even stronger than what Brooks’s theorem gives for vertex coloring: : Vizing’s Theorem: A graph of maximal degree \Delta has edge-chromatic number \Delta or \Delta+1.

The British seized on the concept immediately. Worsaae's earlier and later became Lubbock's palaeo- and neo- in 1865, but alternatively English speakers used Earlier and Later Stone Age, as did Lyell's 1883 edition of Principles of Geology, with older and younger as synonyms. As there is no room for a middle between the comparative adjectives, they were later modified to early and late. The scheme created a problem for further bipartite subdivisions, which would have resulted in such terms as early early Stone Age, but that terminology was avoided by adoption of Geikie's upper and lower Paleolithic.

The Janta Mazdoor Sangh is the largest and the most influential trade union in Dhanbad and in the Bharat Coking Coal Limited. The Sangh was started by the Late Suryadeo Singh and is now headed by Shri Sanjeev Singh. Sanjeev Singh is a member of the JBCCI (Joint Bipartite Committee for Coal Industry), where he represents the workmen and the Hind Mazdoor Sabha. At the behest of their leader, Shri Sanjeev Singh, all the members and supporters of the Janta Mazdoor Sangh wholeheartedly supported the candidate of the BJP from Dhanbad, in the Lok Sabha Election, 2019, thereby ensuring BJP's resounding triumph.

For perfect graphs, it is possible to find a maximum clique in polynomial time, using an algorithm based on semidefinite programming. However, this method is complex and non-combinatorial, and specialized clique-finding algorithms have been developed for many subclasses of perfect graphs. In the complement graphs of bipartite graphs, Kőnig's theorem allows the maximum clique problem to be solved using techniques for matching. In another class of perfect graphs, the permutation graphs, a maximum clique is a longest decreasing subsequence of the permutation defining the graph and can be found using known algorithms for the longest decreasing subsequence problem.

Similarly, in a unit disk graph (with a known geometric representation), there is a polynomial time algorithm for maximum cliques based on applying the algorithm for complements of bipartite graphs to shared neighborhoods of pairs of vertices. The algorithmic problem of finding a maximum clique in a random graph drawn from the Erdős–Rényi model (in which each edge appears with probability , independently from the other edges) was suggested by . Because the maximum clique in a random graph has logarithmic size with high probability, it can be found by a brute force search in expected time . This is a quasi-polynomial time bound.

In 1960, George Sperling became the first to use a partial report paradigm to investigate the bipartite model of VSTM. In Sperling's initial experiments in 1960, observers were presented with a tachistoscopic visual stimulus for a brief period of time (50 ms) consisting of either a 3x3 or 3x4 array of alphanumeric characters such as: :`P Y F G` :`V J S A` :`D H B U` Recall was based on a cue which followed the offset of the stimulus and directed the subject to recall a specific line of letters from the initial display. Memory performance was compared under two conditions: whole report and partial report.

This acts as a protein- interaction motif, similar to those found in other adhesion-related proteins such as focal adhesion kinase (FAK) and vinculin. The remaining carboxy- terminal sequence contains a bipartite Src-binding domain (residues 681–713) able to bind both the SH2 and SH3 domains of Src. p130Cas/BCAR1 can undergo extensive changes in tyrosine phosphorylation that occur predominantly in the 15 YxxP repeats within the substrate domain and represent the major post- translational modification of p130Cas/BCAR1. p130Cas/BCAR1 tyrosine phosphorylation can result from a diverse range of extracellular stimuli, including growth factors, integrin activation, vasoactive hormones and peptides ligands for G-protein coupled receptors.

Infectious bursal disease virus (IBDV) is the best-characterized member of the family Birnaviridae. These viruses have bipartite dsRNA genomes enclosed in single layered icosahedral capsids with T = 13l geometry. IBDV shares functional strategies and structural features with many other icosahedral dsRNA viruses, except that it lacks the T = 1 (or pseudo T = 2) core common to the Reoviridae, Cystoviridae, and Totiviridae. The IBDV capsid protein exhibits structural domains that show homology to those of the capsid proteins of some positive-sense single-stranded RNA viruses, such as the nodaviruses and tetraviruses, as well as the T = 13 capsid shell protein of the Reoviridae.

The primary cycle of four beatsPolyrhythm 6:4 A great deal of African music is built upon a cycle of four main beats. This basic musical period has a bipartite structure; it is made up of two cells, consisting of two beats each. Ladzekpo states: "The first most useful measure scheme consists of four main beats with each main beat measuring off three equal pulsations [] as its distinctive feature … The next most useful measure scheme consists of four main beats with each main beat flavored by measuring off four equal pulsations []." (b: "Main Beat Schemes") The four-beat cycle is a shorter period than what is normally heard in European music.

Lieb-Schultz-Mattis theorem implies that the ground state of the Heisenberg antiferromagnet on a bipartite lattice with isomorphic sublattices, is non-degenerate, i.e., unique, but the gap can be very small.E. Lieb, D. Mattis, Ordering energy levels in interacting spin chains, Journ. Math. Phys. 3, 749–751, (1962) For one-dimensional and quasi-one-dimensional systems of even length and with half-integral spin Affleck and Lieb, generalizing the original result by Lieb, Schultz, and Mattis, proved that the gap \gamma_L in the spectrum above the ground state is bounded above by :\gamma_L\leq c/L, where L is the size of the lattice and c is a constant.

It is NP-complete to determine whether a given directed graph is skew-symmetric, by a result of that it is NP-complete to find a color- reversing involution in a bipartite graph. Such an involution exists if and only if the directed graph given by orienting each edge from one color class to the other is skew-symmetric, so testing skew-symmetry of this directed graph is hard. This complexity does not affect path-finding algorithms for skew-symmetric graphs, because these algorithms assume that the skew-symmetric structure is given as part of the input to the algorithm rather than requiring it to be inferred from the graph alone.

Two different graphs (red) that are duals of the same planar graph (pale blue). Despite being non-isomorphic as graphs, they have isomorphic graphic matroids. A matroid is graphic if and only if its minors do not include any of five forbidden minors: the uniform matroid U{}^2_4, the Fano plane or its dual, or the duals of M(K_5) and M(K_{3,3}) defined from the complete graph K_5 and the complete bipartite graph K_{3,3}.. See in particular section 2.5, "Bond-matroid of a graph", pp. 5–6, section 5.6, "Graphic and co-graphic matroids", pp. 19–20, and section 9, "Graphic matroids", pp.

In information theory, a low-density parity-check (LDPC) code is a linear error correcting code, a method of transmitting a message over a noisy transmission channel.David J.C. MacKay (2003) Information theory, Inference and Learning Algorithms, CUP, , (also available online)Todd K. Moon (2005) Error Correction Coding, Mathematical Methods and Algorithms. Wiley, (Includes code) An LDPC is constructed using a sparse Tanner graph (subclass of the bipartite graph).Amin Shokrollahi (2003) LDPC Codes: An Introduction LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical maximum (the Shannon limit) for a symmetric memoryless channel.

A cycle (or a circuit) in a hypergraph is a cyclic alternating sequence of distinct vertices and hyperedges: (v1, e1, v2, e2, ..., vk, ek, vk+1=v1), where every vertex vi is contained in both ei−1 and ei. The number k is called the length of the cycle. A hypergraph is balanced iff every odd-length cycle C in H has an edge containing at least three vertices of C. Note that in a simple graph all edges contain only two vertices. Hence, a simple graph is balanced iff it contains no odd-length cycles at all, which holds iff it is bipartite.

An n-vertex graph G is pancyclic if, for every k in the range , G contains a cycle of length k. It is node-pancyclic or vertex-pancyclic if, for every vertex v and every k in the same range, it contains a cycle of length k that contains v.. Similarly, it is edge-pancyclic if, for every edge e and every k in the same range, it contains a cycle of length k that contains e. A bipartite graph cannot be pancyclic, because it does not contain any odd-length cycles, but it is said to be bipancyclic if it contains cycles of all even lengths from 4 to n..

The hitting set problem is equivalent to the set cover problem: An instance of set cover can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, elements of the universe represented by vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right- vertices. In the hitting set problem, the objective is to cover the left- vertices using a minimum subset of the right vertices. Converting from one problem to the other is therefore achieved by interchanging the two sets of vertices.

A hypergraph H = (V, E) is called 2-colorable if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge meets both X and Y. Equivalently, the vertices of H can be 2-colored so that no hyperedge is monochromatic. Every bipartite graph G = (X+Y, E) is 2-colorable: each edge contains exactly one vertex of X and one vertex of Y, so e.g. X can be colored blue and Y can be colored yellow and no edge is monochromatic. The property fo 2-colorability was first introduced by Felix Bernstein in the context of set families;.

A non-Desargues (103103) configuration. As a projective configuration, the Desargues configuration has the notation (103103), meaning that each of its ten points is incident to three lines and each of its ten lines is incident to three points. Its ten points can be viewed in a unique way as a pair of mutually inscribed pentagons, or as a self-inscribed decagon . The Desargues graph, a 20-vertex bipartite symmetric cubic graph, is so called because it can be interpreted as the Levi graph of the Desargues configuration, with a vertex for each point and line of the configuration and an edge for every incident point-line pair.

Flock House virus is a small, non-enveloped, icosahedral T=3 insect virus containing a bipartite positive sense ssRNA genome comprising two genes: RNA1 (3.1kb) an RNA2 (1.4kb). RNA1 encodes the RNA-dependent RNA polymerase and also contains a frame-shifted subgenomic RNA 3 (369 nts) that encodes protein B2, responsible for inhibition of RNAi pathways. RNA2 encodes the capsid precursor, alpha, of which 180 copies form the viral capsid of FHV. Upon maturation, alpha undergoes an autocatalytic cleavage in its C-terminus to form beta, forming the main structural capsid component, and gamma, a short hydrophobic peptide required for endosome penetration that remains associated with the viral capsid.

More generally, every planar graph of minimum degree at least three either has an edge of total degree at most 12, or at least 60 edges that (like the edges in the triakis icosahedron) connect vertices of degrees 3 and 10. If all triangular faces of a polyhedron are vertex-disjoint, there exists an edge with smaller total degree, at most eight. Generalizations of the theorem are also known for graph embeddings onto surfaces with higher genus. The theorem cannot be generalized to all planar graphs, as the complete bipartite graphs K_{1,n-1} and K_{2,n-2} have edges with unbounded total degree.

Hinggi Since textiles are the products of Sumbanese women, they are viewed as tangible representations of the female element of the bipartite universe. In Sumba, this male-female dynamic is encapsulated in the notion of the Highest Being, who is both the Father Sun and Mother Moon, as well as the Creator or Weaver of human life. The Sumbanese believe individuals are able to acquire the special powers and qualities of certain creatures when textiles displaying such motifs are worn.Adams M.J., System and Meaning in East Sumba Textile Design: A Study in Traditional Indonesian Art, New Haven, Southeast Asian Cultural report Series 16, Yale University.

The ELMO family are evolutionarily conserved orthologs of the C. elegans protein CED-12. All isoforms contain a series of armadillo repeats, which begin at the N-terminus and extend around two thirds of the way along the protein, as well as a C-terminal proline-rich motif and a central PH domain. They function as part of a protein complex with Dock180-related proteins to form a bipartite guanine nucleotide exchange factor for Rac (a member of the Rho family of small G proteins). The Dock180-ELMO interaction requires the ELMO PH domain and also involves binding of the ELMO proline-rich motif to the Dock180 SH3 domain.

The Birkhoff polytope has n! vertices, one for each permutation on n items. This follows from the Birkhoff–von Neumann theorem, which states that the extreme points of the Birkhoff polytope are the permutation matrices, and therefore that any doubly stochastic matrix may be represented as a convex combination of permutation matrices; this was stated in a 1946 paper by Garrett Birkhoff,. but equivalent results in the languages of projective configurations and of regular bipartite graph matchings, respectively, were shown much earlier in 1894 in Ernst Steinitz's thesis and in 1916 by Dénes Kőnig.. Because all of the vertex coordinates are zero or one, the Birkhoff polytope is an integral polytope.

This article details the qualifying phase for Powerlifting at the 2016 Summer Paralympics. The competition at these Games will comprise a total of 180 athletes coming from their respective NPCs; each has been allowed to enter a maximum of 16 (eight for men, eight for women, and in either case, one per division). 140 will be awarded places based on world rankings in 2016, while 40 are made available to NPCs through a Bipartite Commission Invitation. The top 8 men and top 6 women from the world rankings in each division earn a quota a place, always ensuring that the NPC is subjected to a limit of 1 lifter per division.

By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex- connected, such as the complete bipartite graph , the embedding is not unique, but all embeddings are isomorphic.

The sum of weighted perfect matchings can also be computed by using the Tutte matrix for the adjacency matrix in the last step. Kuratowski's theorem states that : a finite graph is planar if and only if it contains no subgraph homeomorphic to K5 (complete graph on five vertices) or K3,3 (complete bipartite graph on two partitions of size three). Vijay Vazirani generalized the FKT algorithm to graphs that do not contain a subgraph homeomorphic to K3,3. Since counting the number of perfect matchings in a general graph is #P-complete, some restriction on the input graph is required unless FP, the function version of P, is equal to #P.

In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8 it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle W2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a).

Computing tree-depth is computationally hard: the corresponding decision problem is NP-complete.. The problem remains NP-complete for bipartite graphs , as well as for chordal graphs.. On the positive side, tree-depth can be computed in polynomial time on interval graphs,. as well as on permutation, trapezoid, circular-arc, circular permutation graphs, and cocomparability graphs of bounded dimension.. For undirected trees, tree-depth can be computed in linear time.; . give an approximation algorithm for tree-depth with an approximation ratio of O((\log n)^2), based on the fact that tree-depth is always within a logarithmic factor of the treewidth of a graph.

The bipartite NLS is now known to represent the major class of NLS found in cellular nuclear proteins and structural analysis has revealed how the signal is recognized by a receptor (importin α) protein (the structural basis of some monopartite NLSs is also known). Many of the molecular details of nuclear protein import are now known. This was made possible by the demonstration that nuclear protein import is a two-step process; the nuclear protein binds to the nuclear pore complex in a process that does not require energy. This is followed by an energy-dependent translocation of the nuclear protein through the channel of the pore complex.

A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest clique equals the minimum number of colors in a coloring of the subgraph. Perfect graphs include many important graphs classes including bipartite graphs, chordal graphs, and comparability graphs. The complement of a graph has an edge between two vertices if and only if the original graph does not have an edge between the same two vertices. Thus, a clique in the original graph becomes an independent set in the complement and a coloring of the original graph becomes a clique cover of the complement.

As showed, this graph has boxicity exactly n; it is sometimes known as the Roberts graph. This graph is also the 1-skeleton of an n-dimensional cross-polytope; for instance, the graph T(6,3) = K2,2,2 is the octahedral graph, the graph of the regular octahedron. If n couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason it is also called the cocktail party graph. The Turán graph T(n,2) is a complete bipartite graph and, when n is even, a Moore graph.

As a special case of Turán's theorem, for r = 2, one obtains: :Mantel's Theorem. The maximum number of edges in an -vertex triangle-free graph is \lfloor n^2/4 \rfloor. In other words, one must delete nearly half of the edges in to obtain a triangle-free graph. A strengthened form of Mantel's theorem states that any hamiltonian graph with at least n2/4 edges must either be the complete bipartite graph Kn/2,n/2 or it must be pancyclic: not only does it contain a triangle, it must also contain cycles of all other possible lengths up to the number of vertices in the graph .

This protein belongs to the HNF1 homeobox family. It contains 3 functional domains: an N-terminal dimerization domain (residues 1–32), a bipartite DNA-binding motif containing an atypical POU-homeodomain (residues 98–280), and a C-terminal transactivation domain (residues 281–631). There is also a flexible linker (residues 33–97) which connects the dimerization and DNA binding domains. Crystal structures have been solved for the dimerization domain, which forms a four-helix bundle where two α helices are separated by a turn; the DNA-binding motif, which forms a helix-turn-helix structure; and the POU-homeodomain, which is composed of three α helices, contained in the motif.

Compared metavinculin sequences from pig, man, chicken, and frog, and found the insert to be bipartite: the first part variable and the second highly conserved. Both vinculin isoforms co-localize in muscular adhesive structures, such as dense plaques in smooth muscles, intercalated discs in cardiomyocytes, and costameres in skeletal muscles. Metavinculin tail domain has a lower affinity for the head as compared with the vinculin tail. In case of metavinculin, unfurling of the C-terminal hydrophobic hairpin loop of tail domain is impaired by the negative charges of the 68-amino acid insert, thus requiring phospholipid-activated regular isoform of vinculin to fully activate the metavinculin molecule.

Dudeney also published the same puzzle previously, in The Strand Magazine in 1913.. Another early version of the problem involves connecting three houses to three wells.; . It is stated similarly to a different (and solvable) puzzle that also involves three houses and three fountains, with all three fountains and one house touching a rectangular wall; the puzzle again involves making non- crossing connections, but only between three designated pairs of houses and wells or fountains, as in modern numberlink puzzles.. Mathematically, the problem can be formulated in terms of graph drawings of the complete bipartite graph K3,3. This graph makes an early appearance in .. As cited by .

Johann Sebastian Bach composed the church cantata ' (Why would you grieve), 107' in Leipzig for the seventh Sunday after Trinity and first performed on 23 July 1724. The chorale cantata is based on the words of Johann Heermann's hymn in seven stanzas "" (1630). Bach structured the cantata, the seventh work in his chorale cantata cycle, in seven movements: two framing choral movements, a recitative and an unusual sequence of four bipartite arias. He scored the work for three vocal soloists, a four-part choir, and a Baroque chamber ensemble of a horn to reinforce the hymn tune in the outer movements, two transverse flutes, two oboes d'amore, strings and continuo.

K5 (left) and K3,3 (right) as minors of the nonplanar Petersen graph (small colored circles and solid black edges). The minors may be formed by deleting the red vertex and contracting edges within each yellow circle. A clique-sum of two planar graphs and the Wagner graph, forming a K5-free graph. In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K5 (the complete graph on five vertices) nor K3,3 (the utility graph, a complete bipartite graph on six vertices).

Like other members of the tribe Cichorieae, lettuce inflorescences (also known as flower heads or capitula) are composed of multiple florets, each with a modified calyx called a pappus (which becomes the feathery "parachute" of the fruit), a corolla of five petals fused into a ligule or strap, and the reproductive parts. These include fused anthers that form a tube which surrounds a style and bipartite stigma. As the anthers shed pollen, the style elongates to allow the stigmas, now coated with pollen, to emerge from the tube. The ovaries form compressed, obovate (teardrop-shaped) dry fruits that do not open at maturity, measuring 3 to 4 mm long.

A permutation set of an n-by-n matrix X is a set of n entries of X containing exactly one entry from each row and from each column. A theorem by Dénes Kőnig says that:. > Every bistochastic matrix has a permutation-set in which all entries are > positive. The positivity graph of an n-by-n matrix X is a bipartite graph with 2n vertices, in which the vertices on one side are n rows and the vertices on the other side are the n columns, and there is an edge between a row and a column iff the entry at that row and column is positive.

The concept of the line graph of G may naturally be extended to the case where G is a multigraph. In this case, the characterizations of these graphs can be simplified: the characterization in terms of clique partitions no longer needs to prevent two vertices from belonging to the same to cliques, and the characterization by forbidden graphs has seven forbidden graphs instead of nine. However, for multigraphs, there are larger numbers of pairs of non-isomorphic graphs that have the same line graphs. For instance a complete bipartite graph K1,n has the same line graph as the dipole graph and Shannon multigraph with the same number of edges.

To centralize his rule over the emirate (as opposed to the previous bipartite regime with Sheikh Bashir), Emir Bashir proceeded to assume control over legislative and judicial powers by setting up a defined legal code based on the Sharia law of the Ottoman state. Moreover, he transferred jurisdiction over civil and criminal affairs from the mostly Druze muqata'jis to three special qudah (judges; sing. qadi), whom he personally appointed. As such, he assigned a qadi in Deir al-Qamar in Chouf, who mostly oversaw the affairs of the Druze, and two Maronite clergymen who were based in Ghazir or Zouk Mikael in Keserwan and Zgharta in northern Mount Lebanon, respectively.

As their name implies, RBMs are a variant of Boltzmann machines, with the restriction that their neurons must form a bipartite graph: a pair of nodes from each of the two groups of units (commonly referred to as the "visible" and "hidden" units respectively) may have a symmetric connection between them; and there are no connections between nodes within a group. By contrast, "unrestricted" Boltzmann machines may have connections between hidden units. This restriction allows for more efficient training algorithms than are available for the general class of Boltzmann machines, in particular the gradient-based contrastive divergence algorithm.Miguel Á. Carreira-Perpiñán and Geoffrey Hinton (2005).

The first movement (Entrée) is bipartite and juxtaposes several musical ideas: a theme inspired by Susanna's aria Venite inginocchiatevi in Act 2 of Mozart's Le nozze di Figaro, birdsong transcriptions (garden warbler as well as birds of New Zealand like the blue-wattled crow, the bush canary and the kakapo), a call and response of short melodic cells, a section for wind machine, strings and cymbal and two conclusive chords. That sequence is then repeated and amplified in the second part.Griffiths, Paul (1995), Concert à quatre (Deutsche Grammophon), English liner notes. The second movement is an orchestral transcription of Messiaen's own Vocalise of 1935.

A cut of an undirected graph is a partition of the vertices into two nonempty subsets, the sides of the cut. The subset of edges that have one endpoint in each side is called a cut-set. When a cut-set forms a complete bipartite graph, its cut is called a split. Thus, a split can be described as a partition of the vertices of the graph into two subsets and , such that every neighbor of in is adjacent to every neighbor of in .. A cut or split is trivial when one of its two sides has only one vertex in it; every trivial cut is a split.

The slots cannot be transferred between gender; any unused slots are re allocated via the Bipartite Commission Invitation Allocation method. Each NPC may enter up to three athletes in each individual event, one team of up to six athletes (but no less than four) in each relay event, and up to six athletes in each marathon event as long as no more than three athletes are competing in the marathon as their sole event. There is no limit on how many events an individual athlete may be entered in, as long as they have achieved the 'B' qualifying standard in that event between 15 October 2014 and 14 August 2016.

Therefore, we have constructed an antichain and a partition into chains with the same cardinality. To prove Kőnig's theorem from Dilworth's theorem, for a bipartite graph G = (U,V,E), form a partial order on the vertices of G in which u < v exactly when u is in U, v is in V, and there exists an edge in E from u to v. By Dilworth's theorem, there exists an antichain A and a partition into chains P both of which have the same size. But the only nontrivial chains in the partial order are pairs of elements corresponding to the edges in the graph, so the nontrivial chains in P form a matching in the graph.

The constraint hypergraph of a constraint satisfaction problem is a hypergraph in which the vertices correspond to the variables, and the hyperedges correspond to the constraints. A set of vertices forms a hyperedge if the corresponding variables are those occurring in some constraint. A simple way to represent the constraint hypergraph is by using a classical graph with the following properties: # Vertices correspond either to variables or to constraints, # an edge can only connect a variable-vertex to a constraint-vertex, and # there is an edge between a variable-vertex and a constraint-vertex if and only if the corresponding variable occurs in the corresponding constraint. Properties 1 and 2 define a bipartite graph.

In Ayt Ayache the Arabic numerals are only used for counting in order and for production of higher numbers when combined with the tens, see All higher cardinals are borrowed from Arabic, consistent with the linguistic universals that the numbers 1–3 are much more likely to be retained, and that a borrowed number generally implies that numbers greater than it are also borrowed. The retention of one is also motivated by the fact that Berber languages near-universally use unity as a determiner. Central Atlas Tamazight uses a bipartite negative construction (e.g. /uriffiɣ ʃa/ 'he did not go out') which apparently was modeled after proximate Arabic varieties, in a common development known as Jespersen's Cycle.

A slight modification of the above game, and the related graph-theoretic problem, makes solving the game NP-hard. The modified game has the Rabin acceptance condition. Specifically, in the above bipartite graph scenario, the problem now is to determine if there is a choice function selecting a single out-going edge from each vertex of V0, such that the resulting subgraph has the property that in each cycle (and hence each strongly connected component) it is the case that there exists an i and a node with color 2i, and no node with color 2i + 1... Note that as opposed to parity games, this game is no longer symmetric with respect to players 0 and 1\.

Therefore, the sum of the size of the largest independent set \alpha(G) and the size of a minimum vertex cover \beta(G) is equal to the number of vertices in the graph. A vertex coloring of a graph G corresponds to a partition of its vertex set into independent subsets. Hence the minimal number of colors needed in a vertex coloring, the chromatic number \chi(G), is at least the quotient of the number of vertices in G and the independent number \alpha(G). In a bipartite graph with no isolated vertices, the number of vertices in a maximum independent set equals the number of edges in a minimum edge covering; this is Kőnig's theorem.

Let G = (U,V,E) be a bipartite graph. One may define a partition matroid MU on the ground set E, in which a set of edges is independent if no two of the edges have the same endpoint in U. Similarly one may define a matroid MV in which a set of edges is independent if no two of the edges have the same endpoint in V. Any set of edges that is independent in both MU and MV has the property that no two of its edges share an endpoint; that is, it is a matching. Thus, the largest common independent set of MU and MV is a maximum matching in G.

The Moser spindle, a planar Laman graph drawn as a pointed pseudotriangulation The complete bipartite graph K3,3, a non-planar Laman graph In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k − 3 edges, and such that the whole graph has exactly 2n − 3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures.. This characterization, however, had already been discovered in 1927 by Hilda Geiringer..

In 1993,Brouwer A. E.; Dejter I. J.; Thomassen C. "Highly symmetric subgraphs of hypercubes", J. Algebraic Combinat. 2, 22-25, 1993 Brouwer, Dejter and Thomassen described an undirected, bipartite graph with 112 vertices and 168 edges, (semi-symmetric, that is edge-transitive but not vertex-transitive, cubic graph with diameter 8, radius 7, chromatic number 2, chromatic index 3, girth 10, with exactly 168 cycles of length 10 and 168 cycles of length 12), known since 2002 as the Ljubljana graph. They also established that the Dejter graph,Klin M.; Lauri J.; Ziv-Av M. "Links between two semisymmetric graphs on 112 vertices through the lens of association schemes", Jour. Symbolic Comput., 47–10, 2012, 1175–1191.

By symmetry, each edge of the K6 belongs to three perfect matchings. Incidentally, this partitioning of vertices into edge-vertices and matching-vertices shows that the Tutte-Coxeter graph is bipartite. Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a symmetric graph; it has a group of 1440 automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The inner automorphisms of this group correspond to permuting the six vertices of the K6 graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set.

As distance-regular graphs, they are uniquely defined by their intersection array: no other distance-regular graphs can have the same parameters as an odd graph.. However, despite their high degree of symmetry, the odd graphs On for n > 2 are never Cayley graphs.. Because odd graphs are regular and edge-transitive, their vertex connectivity equals their degree, n. Odd graphs with n > 3 have girth six; however, although they are not bipartite graphs, their odd cycles are much longer. Specifically, the odd graph On has odd girth 2n − 1\. If a n-regular graph has diameter n − 1 and odd girth 2n − 1, and has only n distinct eigenvalues, it must be distance-regular.

An undirected graph G is Hamiltonian if it contains a cycle that touches each of its vertices exactly once. It is 2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex- connected graph is Hamiltonian; counterexamples include the Petersen graph and the complete bipartite graph K2,3. The square of G is a graph G2 that has the same vertex set as G, and in which two vertices are adjacent if and only if they have distance at most two in G. Fleischner's theorem states that the square of a finite 2-vertex-connected graph with at least three vertices must always be Hamiltonian.

In fact, the problem has a kernel of size linear in k, and running times that are exponential in and cubic in n may be obtained by applying dynamic programming to a branch-decomposition of the kernel. More generally, the dominating set problem and many variants of the problem are fixed-parameter tractable when parameterized by both the size of the dominating set and the size of the smallest forbidden complete bipartite subgraph; that is, the problem is FPT on biclique-free graphs, a very general class of sparse graphs that includes the planar graphs. The complementary set to a dominating set, a nonblocker, can be found by a fixed-parameter algorithm on any graph.

Qualification will be largely based on the world ranking list prepared by International Blind Sports Federation as of May 30, 2016. A total of 114 athletes will directly qualify through the ranking with only the top 9 men or top 6 women in each division, ensuring that each NPC is subjected to a limit of one judoka per division. In addition, a quota place will be reserved in each weight category for the host nation, and a place will also be reserved for the NPC of the winners of each weight division at the 2014 IBSA World Championships. finally, the Bipartite Commission will also make one invite per weight division, to complete the field.

This article details the qualifying phase for Judo at the 2016 Summer Paralympics. The competition at these Games will comprise a total of 132 athletes coming from their respective NPCs; each has been allowed to enter a maximum of 13 (seven for men, six for women, and in either case, one per division). Host nation Brazil has reserved a spot in each of all 13 events, while 13 are made available to NPCs through a Bipartite Commission Invitation. The remaining judoka must undergo a qualifying process to earn a spot for the Games through the world ranking list prepared by IBSA that begins on December 31, 2014 and then concludes one year later on the same date.

The minimum number of induced matchings into which the edges of a graph can be partitioned is called its strong chromatic index, by analogy with the chromatic index of the graph, the minimum number of matchings into which its edges can be partitioned. It equals the chromatic number of the square of the line graph. Brooks' theorem, applied to the square of the line graph, shows that the strong chromatic index is at most quadratic in the maximum degree of the given graph, but better constant factors in the quadratic bound can be obtained by other methods. The Ruzsa–Szemerédi problem concerns the edge density of balanced bipartite graphs with linear strong chromatic index.

Pisanski’s research interests span several areas of discrete and computational mathematics, including combinatorial configurations, abstract polytopes, maps on surfaces, chemical graph theory, and the history of mathematics and science. In 1980 he calculated the genus of the Cartesian product of any pair of connected, bipartite, d-valent graphs using a method that was later called the White–Pisanski method.J.L. Gross and T.W. Tucker, Topological graph theory, Wiley Interscience, 1987 In 1982 Vladimir Batagelj and Pisanski proved that the Cartesian product of a tree and a cycle is Hamiltonian if and only if no degree of the tree exceeds the length of the cycle. They also proposed a conjecture concerning cyclic Hamiltonicity of graphs.

During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory, pushing wagon loads of bricks from kilns to storage sites. The factory had tracks from each kiln to each storage site, and the wagons were harder to push at the points where tracks crossed each other. Turán was inspired by this situation to ask how the factory might be redesigned to minimize the number of crossings between these tracks. Mathematically, this problem can be formalized as asking for a graph drawing of a complete bipartite graph, whose vertices represent kilns and storage sites, and whose edges represent the tracks from each kiln to each storage site.

Every bipartite graph is of class 1, and almost all random graphs are of class 1.. However, it is NP-complete to determine whether an arbitrary graph is of class 1.. proved that planar graphs of maximum degree at least eight are of class one and conjectured that the same is true for planar graphs of maximum degree seven or six. On the other hand, there exist planar graphs of maximum degree ranging from two through five that are of class two. The conjecture has since been proven for graphs of maximum degree seven.. Bridgeless planar cubic graphs are all of class 1; this is an equivalent form of the four color theorem.; .

For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G. The list coloring number ch(G) satisfies the following properties. # ch(G) ≥ χ(G). A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring. # ch(G) cannot be bounded in terms of chromatic number in general, that is, there is no function f such that ch(G) ≤ f(χ(G)) holds for every graph G. In particular, as the complete bipartite graph examples show, there exist graphs with χ(G) = 2 but with ch(G) arbitrarily large.

In matroids, a non-separating circuit is a circuit of the matroid (that is, a minimal dependent set) such that deleting the circuit leaves a smaller matroid that is connected (that is, that cannot be written as a direct sum of matroids).. These are analogous to peripheral cycles, but not the same even in graphic matroids (the matroids whose circuits are the simple cycles of a graph). For example, in the complete bipartite graph K_{2,3}, every cycle is peripheral (it has only one bridge, a two-edge path) but the graphic matroid formed by this bridge is not connected, so no circuit of the graphic matroid of K_{2,3} is non-separating.

The main text of the book has two parts, on the crossing number as traditionally defined and on variations of the crossing number, followed by two appendices providing background material on topological graph theory and computational complexity theory. After introducing the problem, the first chapter studies the crossing numbers of complete graphs (including Hill's conjectured formula for these numbers) and complete bipartite graphs (Turán's brick factory problem and the Zarankiewicz crossing number conjecture), again giving a conjectured formula). It also includes the crossing number inequality, and the Hanani–Tutte theorem on the parity of crossings. The second chapter concerns other special classes of graphs including graph products (especially products of cycle graphs) and hypercube graphs.

There have also been several results relating coloring to minimum degree in triangle-free graphs. proved that any n-vertex triangle-free graph in which each vertex has more than 2n/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2n/5 neighbors per vertex. Motivated by this result, conjectured that any n-vertex triangle-free graph in which each vertex has at least n/3 neighbors can be colored with only three colors; however, disproved this conjecture by finding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent set of a carefully chosen size.

The critic Joseph Bédier (1864–1938), who had worked with stemmatics, launched an attack on that method in 1928. He surveyed editions of medieval French texts that were produced with the stemmatic method, and found that textual critics tended overwhelmingly to produce bifid trees, divided into just two branches. He concluded that this outcome was unlikely to have occurred by chance, and that therefore, the method was tending to produce bipartite stemmas regardless of the actual history of the witnesses. He suspected that editors tended to favor trees with two branches, as this would maximize the opportunities for editorial judgment (as there would be no third branch to "break the tie" whenever the witnesses disagreed).

Pavol Hell and Jaroslav Nešetřil proved that, for undirected graphs, no other case is tractable: : Hell–Nešetřil theorem (1990): The H-coloring problem is in P when H is bipartite and NP-complete otherwise. This is also known as the dichotomy theorem for (undirected) graph homomorphisms, since it divides H-coloring problems into NP-complete or P problems, with no intermediate cases. For directed graphs, the situation is more complicated and in fact equivalent to the much more general question of characterizing the complexity of constraint satisfaction problems. It turns out that H-coloring problems for directed graphs are just as general and as diverse as CSPs with any other kinds of constraints.

A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. The line graph of the complete graph Kn is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph KGn,2. Triangular graphs are characterized by their spectra, except for n = 8.. See in particular Proposition 8, p. 262. They may also be characterized (again with the exception of K8) as the strongly regular graphs with parameters srg(n(n − 1)/2, 2(n − 2), n − 2, 4).

The Upper Rhine Conference (officially known as the Franco-German-Swiss Conference of the Upper Rhine) provides the institutional framework for cross- border cooperation in the Upper Rhine region. It is the successor organization to the two regional commissions (bipartite regional commission for the northern and tripartite regional commission for the southern Upper Rhine region) which derived from the 1975 Upper Rhine agreement between Germany, France and Switzerland,The "Bonn Accord", a tripartite agreement between the Federal Republic of Germany, the French Republic and the Swiss Confederation on the formation of a commission for studying and resolving regional cross- border issues, 22 October 1975 which were established to work under the auspices of the Franco-German-Swiss Intergovernmental Commission.

One wants a subset S of the vertex set such that the number of edges between S and the complementary subset is as large as possible. Equivalently, one wants a bipartite subgraph of the graph with as many edges as possible. There is a more general version of the problem called weighted Max-Cut, where each edge is associated with a real number, its weight, and the objective is to maximize the total weight of the edges between S and its complement rather than the number of the edges. The weighted Max-Cut problem allowing both positive and negative weights can be trivially transformed into a weighted minimum cut problem by flipping the sign in all weights.

The Apollonian networks do not form a family of graphs that is closed under the operation of taking graph minors, as removing edges but not vertices from an Apollonian network produces a graph that is not an Apollonian network. However, the planar partial 3-trees, subgraphs of Apollonian networks, are minor-closed. Therefore, according to the Robertson–Seymour theorem, they can be characterized by a finite number of forbidden minors. The minimal forbidden minors for the planar partial 3-trees are the four minimal graphs among the forbidden minors for the planar graphs and the partial 3-trees: the complete graph , the complete bipartite graph , the graph of the octahedron, and the graph of the pentagonal prism.

Determining if a graph can be colored with 2 colors is equivalent to determining whether or not the graph is bipartite, and thus computable in linear time using breadth- first search or depth-first search. More generally, the chromatic number and a corresponding coloring of perfect graphs can be computed in polynomial time using semidefinite programming. Closed formulas for chromatic polynomial are known for many classes of graphs, such as forests, chordal graphs, cycles, wheels, and ladders, so these can be evaluated in polynomial time. If the graph is planar and has low branch-width (or is nonplanar but with a known branch decomposition), then it can be solved in polynomial time using dynamic programming.

In graph theory, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a group. It is formally identical to a gain graph, but it is generally used in topological graph theory as a concise way to specify another graph called the derived graph of the voltage graph. Typical choices of the groups used for voltage graphs include the two-element group ℤ2 (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤd (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space),; ; . and finite cyclic groups ℤn for n > 2\.

Moreover, information contained by edges whose "target" nodes are of degree 1 in the original network will be lost in the projection, which can have grievous consequences in some real networks with a lot of independent edge sets. To overcome these shortcomings, Zhou et al. has proposed a weighting method that is based on assuming that a certain amount of resource is associated with each node in the projection, and the directional weight w_ij represents the proportion of the resource node j would like to distribute to node i. Resource allocation is based on the bipartite graph, involves equal distribution across neighbors and consists of two steps: first from the projected set to the non-projected set, and then back.

At the 1899 convention in Pittsburgh, Pennsylvania, the union's name was officially changed to the International Brotherhood of Electrical Workers. The union went through lean times in its early years, then struggled through six years of schism during the 1910s, when two rival groups each claimed to be the duly elected leaders of the union. In 1919, as many employers were trying to drive unions out of the workplace through a national open shop campaign, the union agreed to form the Council on Industrial Relations, a bipartite body made up of equal numbers of management and union representatives with the power to resolve any collective bargaining disputes. That body still functions today, and has largely resolved strikes in the IBEW's jurisdiction in the construction industry.

A -map graph is a map graph derived from a set of regions in which at most regions meet at any point. Equivalently, it is the half-square of a planar bipartite graph in which the vertex set (the side of the bipartition not used to induce the half-square) has maximum degree . A 3-map graph is a planar graph, and every planar graph can be represented as a 3-map graph. Every 4-map graph is a 1-planar graph, a graph that can be drawn with at most one crossing per edge, and every optimal 1-planar graph (a graph formed from a planar quadrangulation by adding two crossing diagonals to every quadrilateral face) is a 4-map graph.

The "vagina" of monotremes is better understood as a "urogenital sinus". The uterine systems of placental mammals can vary between a duplex, were there are two uteri and cervices which open into the vagina, a bipartite, were two uterine horns have a single cervix that connects to the vagina, a bicornuate, which consists where two uterine horns that are connected distally but separate medially creating a Y-shape, and a simplex, which has a single uterus. Matschie's tree-kangaroo with young in pouch The ancestral condition for mammal reproduction is the birthing of relatively undeveloped, either through direct vivipary or a short period as soft-shelled eggs. This is likely due to the fact that the torso could not expand due to the presence of epipubic bones.

The 12-vertex crown graph, the intersection graph of the lines of the double six A generic cubic surface contains 27 lines, among which can be found 36 Schläfli double six configurations. The set of 15 lines complementary to a double six, together with the 15 tangent planes through triples of these lines, has the incidence pattern of another configuration, the Cremona–Richmond configuration. The intersection graph of the twelve lines of the double six configuration is a twelve-vertex crown graph, a bipartite graph in which each vertex is adjacent to five out of the six vertices of the opposite color. The Levi graph of the double six may be obtained by replacing each edge of the crown graph by a two-edge path.

In combinatorial optimization, the matroid intersection problem is to find a largest common independent set in two matroids over the same ground set. If the elements of the matroid are assigned real weights, the weighted matroid intersection problem is to find a common independent set with the maximum possible weight. These problems generalize many problems in combinatorial optimization including finding maximum matchings and maximum weight matchings in bipartite graphs and finding arborescences in directed graphs. The matroid intersection theorem, due to Jack Edmonds, says that there is always a simple upper bound certificate, consisting of a partitioning of the ground set amongst the two matroids, whose value (sum of respective ranks) equals the size of a maximum common independent set.

Mainstream Greek linguistics separates the Greek dialects into two large genetic groups, one including Doric Greek and the other including both Arcadocypriot and Ionic Greek. But alternative approaches proposing three groups are not uncommon; Thumb and Kieckers (1932) propose three groups, classifying Ionic as genetically just as separate from Arcadocypriot as from Doric. Like a few other linguists (Vladimir Georgiev, C. Rhuijgh, P. Léveque, etc.), the bipartite classification is known as the "Risch–Chadwick theory", named after its two famous proponents, Ernst Risch and John Chadwick. The "Proto-Ionians" first appear in the work of Ernst Curtius (1887), who believed that the Attic- Ionic dialect group was due to an "Ionicization" of Attica by immigration from Ionia in historical times.

If a graph has a Hamiltonian path, then that path (rooted at one of its endpoints) is also a Trémaux tree. The undirected graphs for which every Trémaux tree has this form are the cycle graphs, complete graphs, and balanced complete bipartite graphs.. Trémaux trees are closely related to the concept of tree-depth. The tree-depth of a graph G can be defined as the smallest number d such that G can be embedded as a subgraph of a graph H that has a Trémaux tree T of depth d. Bounded tree- depth, in a family of graphs, is equivalent to the existence of a path that cannot occur as a graph minor of the graphs in the family.

Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains the proof of Kőnig's theorem for regular bipartite graphs, phrased in the language of configurations. In 1910 Steinitz published the very influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields, Crelle's Journal (1910), 167–309). In this paper he axiomatically studies the properties of fields and defines important concepts like prime field, perfect field and the transcendence degree of a field extension.

Determining whether there is an edge dominating set of a given size for a given graph is an NP-complete problem (and therefore finding a minimum edge dominating set is an NP-hard problem). show that the problem is NP-complete even in the case of a bipartite graph with maximum degree 3, and also in the case of a planar graph with maximum degree 3. There is a simple polynomial-time approximation algorithm with approximation factor 2: find any maximal matching. A maximal matching is an edge dominating set; furthermore, a maximal matching M can be at worst 2 times as large as a smallest maximal matching, and a smallest maximal matching has the same size as the smallest edge dominating set.

For these graphs, a convex (but not necessarily strictly convex) drawing can be found within a grid whose length on each side is linear in the number of vertices of the graph, in linear time. However, strictly convex drawings may require larger grids; for instance, for any polyhedron such as a pyramid in which one face has a linear number of vertices, a strictly convex drawing of its graph requires a grid of cubic area. A linear-time algorithm can find strictly convex drawings of polyhedral graphs in a grid whose length on each side is quadratic. Convex but not strictly convex drawing of the complete bipartite graph K_{2,3} Other graphs that are not polyhedral can also have convex drawings, or strictly convex drawings.

Some graphs, such as the complete bipartite graph K_{2,3}, have convex drawings but not strictly convex drawings. A combinatorial characterization for the graphs with convex drawings is known, and they can be recognized in linear time, but the grid dimensions needed for their drawings and an efficient algorithm for constructing small convex grid drawings of these graphs are not known in all cases. Convex drawings should be distinguished from convex embeddings, in which each vertex is required to lie within the convex hull of its neighboring vertices. Convex embeddings can exist in dimensions other than two, do not require their graph to be planar, and even for planar embeddings of planar graphs do not necessarily force the outer face to be convex.

Altar The original altar, depicting Mary(am) of Nazareth, was translated in 1924 to St. Nicholas' Church in Berlin's central borough of Mitte and is now exhibited in the Märkisches Museum. In the same year Tabor Church received in return a wooden altar (of the last quarter of the 15th century) from the village church in Berlin-Wartenberg, showing the carved sculptures of Mary(am) of Nazareth with the infant Jesus in the central field, flanked by two bipartite folding flaps with sculptures of Saints. The pulpit dates to the early 17th century and is decorated with diamond-styled Herms pilasters. Organ and parapet of the loft The christening bowl of 1671 shows the arms of the von Röbel family, who donated it.

Han-era Chinese believed that a person had two souls, the hun and po. The spirit-soul (hun 魂) was believed to travel to the paradise of the immortals (xian 仙) while the body-soul (po 魄) remained on earth in its proper resting place so long as measures were taken to prevent it from wandering to the netherworld. The body- soul could allegedly utilize items placed in the tomb of the deceased, such as domestic wares, clothes, food and utensils, and even money in the form of clay replicas. It was believed that the bipartite souls could also be temporarily reunited in a ceremony called "summoning the hun to return to the po" (zhao hun fu po 招魂復魄).Csikszentmihalyi (2006), 140-141.

At the same time, the traditional notion of "unfree" dependents and the distinction between "unfree" and "free" tenants was eroded as the concept of serfdom (see also History of serfdom) came to dominate.Wickham, 538. From the mid-8th century on, particularly in the north, the relationship between peasants and the land became increasingly characterized by the extension of the new "bipartite estate" system (manors, manorialism), in which peasants (who were bound to the land) held tenant holdings from a lord or monastery (for which they paid rent), but were also required to work the lord's own "demesne"; in the north, some of these estates could be quite substantial.Wickham, 534-5. This system remained a standard part of lord-tenant relations into the 12th century.

Meshulam's game is a game played on a graph G, that can be used to calculate a lower bound on the homological connectivity of the independence complex of G. The matching complex of a graph G, denoted M(G), is an abstract simplicial complex of the matchings in G. It is the independence complex of the line graph of G. The (m,n)-chessboard complex is the matching complex on the complete bipartite graph Km,n. It is the abstract simplicial complex of all sets of positions on an m-by-n chessboard, on which it is possible to put rooks without each of them threatening the other. The clique complex of G is the independence complex of the complement graph of G.

In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by . The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible, that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when G is a bipartite graph, whose vertices V are partitioned between left vertices in X and right vertices in Y, and edges in E always connect a left vertex to a right vertex.

186-187, 1990 Such graphs are called semi-symmetric graphs and were first studied by Folkman in 1967 who discovered the graph on 20 vertices that now is named after him. As a semi-symmetric graph, the Folkman graph is bipartite, and its automorphism group acts transitively on each of the two vertex sets of the bipartition. In the diagram below indicating the chromatic number of the graph, the green vertices can not be mapped to red ones by any automorphism, but any red vertex can be mapped on any other red vertex and any green vertex can be mapped on any other green vertex. The characteristic polynomial of the Folkman graph is (x-4) x^{10} (x+4) (x^2-6)^4.

He asserts that "any quantum computation with pure states can be efficiently simulated with a classical computer provided the amount of entanglement involved is sufficiently restricted" . This happens to be the case with generic Hamiltonians displaying local interactions, as for example, Hubbard-like Hamiltonians. The method exhibits a low-degree polynomial behavior in the increase of computational time with respect to the amount of entanglement present in the system. The algorithm is based on a scheme that exploits the fact that in these one- dimensional systems the eigenvalues of the reduced density matrix on a bipartite split of the system are exponentially decaying, thus allowing us to work in a re-sized space spanned by the eigenvectors corresponding to the eigenvalues we selected.

Secondary technical schools were never widely implemented and for 20 years there was a virtual bipartite system which saw fierce competition for the available grammar school places, which varied between 15% and 25% of total secondary places, depending on location. In 1970 Margaret Thatcher, the Secretary of State for Education in the new Conservative government, ended the compulsion on local authorities to convert, however, many local authorities were so far down the path that it would have been prohibitively expensive to attempt to reverse the process, and more comprehensive schools were established under Thatcher than any other education secretary. By 1975 the majority of local authorities in England and Wales had abandoned the 11-Plus examination and moved to a comprehensive system.

Externally, the authoritarian Regency of Prince Pavle was trying to balance popular pro-French and pro- British sentiments with a need to maintain good relations with the country's Fascist Italian and Nazi German neighbours. Internally, political life was becoming increasingly dominated by Serb-Croat hostility; in 1939, a vain attempt was made to defuse this by imposing a bipartite federal system, under which most of Bosnia became Croatian territory. In late 1939 or early 1940, Skender Kulenović was expelled from the KPJ for having refused to sign an open letter criticising the government and advocating autonomy for Bosnia and Herzegovina – a decision which prevented him from publishing in many of the journals he had worked with until then. In 1940 he married his first wife, Ana Prokop.

When the entries of A are nonnegative, the permanent can be computed approximately in probabilistic polynomial time, up to an error of εM, where M is the value of the permanent and ε > 0 is arbitrary. In other words, there exists a fully polynomial-time randomized approximation scheme (FPRAS) (). The most difficult step in the computation is the construction of an algorithm to sample almost uniformly from the set of all perfect matchings in a given bipartite graph: in other words, a fully polynomial almost uniform sampler (FPAUS). This can be done using a Markov chain Monte Carlo algorithm that uses a Metropolis rule to define and run a Markov chain whose distribution is close to uniform, and whose mixing time is polynomial.

The Northern Black Forest is bounded in the north by a line from Karlsruhe to Pforzheim and, in the south, by a line running from the Rench valley to Freudenstadt. Its northern boundary largely coincides with the emergence of the extensively forested bunter sandstone strata from the arable region of the Kraichgau; its southern boundary with the Central Black Forest (or, in the case of a bipartite division, the Southern Black Forest) varies depending on the definition or natural regional division used (see also Black Forest). Earlier, the Northern Black Forest was the entire northern half of the mountain range as far as the line of the Kinzig valley, which divides the Black Forest east of Lahr. To the west it is bounded by the Upper Rhine Plain, to the east by the Gäu landscapes.

Then, a Hamiltonian path exists if and only if there is a set of n − 1 elements in the intersection of three matroids on the edge set of the graph: two partition matroids ensuring that the in-degree and out-degree of the selected edge set are both at most one, and the graphic matroid of the undirected graph formed by forgetting the edge orientations in G, ensuring that the selected edge set has no cycles . Another computational problem on matroids, the matroid parity problem, was formulated by as a common generalization of matroid intersection and non-bipartite graph matching. However, although it can be solved in polynomial time for linear matroids, it is NP-hard for other matroids, and requires exponential time in the matroid oracle model .

As a result, the tripartite system was in effect a bipartite system in which children who passed the eleven-plus examination were sent to grammar schools and those who failed the test went to secondary modern schools. At a secondary modern school, pupils would receive training in a wide range of simple, practical skills. The purpose of this education was to mainly focus on training in basic subjects, such as arithmetic, mechanical skills such as woodworking, and domestic skills, such as cookery. In an age before the advent of the National Curriculum, the specific subjects taught were chosen by the individual schools, but the curriculum at the Frank Montgomery School in Kent was stated as including "practical education, such as cookery, laundry, gardening, woodwork, metalwork and practical geography".

Nestedness is a measure of structure in an ecological system, usually applied to species-sites systems (describing the distribution of species across locations), or species-species interaction networks (describing the interactions between species, usually as bipartite networks such as hosts- parasites, plants-pollinators, etc.). A system (usually represented as a matrix) is said to be nested when the elements that have a few items in them (locations with few species, species with few interactions) have a subset of the items of elements with more items. Imagine a series of islands that are ordered by their distance from the mainland. If the mainland has all species, the first island has a subset of mainland's species, the second island has a subset of the first island's species, and so forth, then this system is perfectly nested.

In the case of bipartite graphs or multigraphs with maximum degree , the optimal number of colors is exactly . showed that an optimal edge coloring of these graphs can be found in the near-linear time bound , where is the number of edges in the graph; simpler, but somewhat slower, algorithms are described by and . The algorithm of begins by making the input graph regular, without increasing its degree or significantly increasing its size, by merging pairs of vertices that belong to the same side of the bipartition and then adding a small number of additional vertices and edges. Then, if the degree is odd, Alon finds a single perfect matching in near-linear time, assigns it a color, and removes it from the graph, causing the degree to become even.

In the theory of quantum communication, the entanglement-assisted classical capacity of a quantum channel is the highest rate at which classical information can be transmitted from a sender to receiver when they share an unlimited amount of noiseless entanglement. It is given by the quantum mutual information of the channel, which is the input-output quantum mutual information maximized over all pure bipartite quantum states with one system transmitted through the channel. This formula is the natural generalization of Shannon's noisy channel coding theorem, in the sense that this formula is equal to the capacity, and there is no need to regularize it. An additional feature that it shares with Shannon's formula is that a noiseless classical or quantum feedback channel cannot increase the entanglement-assisted classical capacity.

Thus, in this case, the perfect graph theorem implies Kőnig's theorem that the size of a maximum independent set in a bipartite graph is also n − M,, later rediscovered by . a result that was a major inspiration for Berge's formulation of the theory of perfect graphs. Mirsky's theorem characterizing the height of a partially ordered set in terms of partitions into antichains can be formulated as the perfection of the comparability graph of the partially ordered set, and Dilworth's theorem characterizing the width of a partially ordered set in terms of partitions into chains can be formulated as the perfection of the complements of these graphs. Thus, the perfect graph theorem can be used to prove Dilworth's theorem from the (much easier) proof of Mirsky's theorem, or vice versa.

In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphs, there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs. A first definition of acyclicity for hypergraphs was given by Claude Berge: a hypergraph is Berge- acyclic if its incidence graph (the bipartite graph defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair v eq v' of vertices and some pair f eq f' of hyperedges such that v, v' \in f and v, v' \in f', then it is Berge-cyclic. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.

Depending on the level of acceptance of rejection of certain traditions, the interpretation of the Koran can be changed immensely, from the Qur'anists who reject the ahadith, to the Salafi, or ahl al-hadith, who hold the entirety of the traditional collections in great reverence. Traditional Islam views the world as bipartite, consisting of the House of Islam, that is, where people live under the Sharia; and the House of War, that is, where the people do not live under Sharia, which must be proselytized using whatever resources available, including, in some traditionalist and conservative interpretations,Ibn Kathir's Tafsir al-Qur'an al-Aziz the use of violence, as holy struggle in the path of God,Sayyid Qutb Milestones to either convert its inhabitants to Islam, or to rule them under the Shariah (cf. dhimmi).

Structure of typical chromists compared with plant cell (left) Members of Chromista are single-celled and multicellular eukaryotes having basically either or both features: #plastid(s) that contain chlorophyll c and lies within an extra (periplastid) membrane in the lumen of the rough endoplasmic reticulum (typically within the perinuclear cisterna); #cilia with tripartite or bipartite rigid tubular hairs. Even though the kingdom includes diverse organisms from algae to malarial parasites (Plasmodium), they are genetically related and are believed to have evolved from a common ancestor with all other eukaryotes but in an independent line of evolution. As a result of evolution, many have retained their plastids and cilia, while some have lost them. Molecular evidences indicate that the plastids in chromists were derived from red algae through secondary symbiogenesis in a single event.

Stylistically, there are many differences between these works by Telemann and Johann Sebastian Bach's Passions, there are however more similarities with the Passions of C.P.E. Bach. The Telemann Passions were (unlike J. S. Bach's Leipzig Passions) not written for and used in the context of a separate Good Friday Vespers liturgical service, but rather in the regular church services for the five main churches in Hamburg for the Sundays of Lent (except for Oculi Sunday). In deference to Ulrich Leisinger, who states in the Passions Preface in the Carl Philipp Emanuel Bach Complete Works edition: and quotes (he states) pages 656–657 of the 4th Vorrath (Volume) of Johann Mattheson's Plus Ultra, ein Stückwerk von neuer und mancherley Art. Hamburg never really adapted the bipartite division of Passion settings (Part 1 being before the Sermon and Part 2 after it).

United Nations Security Council resolution 626, adopted unanimously on 20 December 1988, after noting an agreement between Angola and Cuba regarding the withdrawal of Cuban troops from Angola and considering a report by the Secretary-General, the Council endorsed the report and established the United Nations Angola Verification Mission I for a period of thirty-one months. The Council decided the mission would enter into force once the tripartite accord between Angola, Cuba and South Africa had been signed as well as the agreement between Angola and Cuba, requesting the Secretary-General to report to the Council immediately after the agreement was signed. On 22 December 1988, both bipartite and tripartite agreements were signed in New York City, which helped pave the way for the independence of Namibia and the withdrawal of 50,000 Cuban troops from Angola.

Bidimensionality theory has been used to obtain polynomial-time approximation schemes for many bidimensional problems. If a minor (contraction) bidimensional problem has several additional properties then the problem poses efficient polynomial-time approximation schemes on (apex) minor-free graphs. In particular, by making use of bidimensionality, it was shown that feedback vertex set, vertex cover, connected vertex cover, cycle packing, diamond hitting set, maximum induced forest, maximum induced bipartite subgraph and maximum induced planar subgraph admit an EPTAS on H-minor-free graphs. Edge dominating set, dominating set, r-dominating set, connected dominating set, r-scattered set, minimum maximal matching, independent set, maximum full-degree spanning tree, maximum induced at most d-degree subgraph, maximum internal spanning tree, induced matching, triangle packing, partial r-dominating set and partial vertex cover admit an EPTAS on apex-minor-free graphs.

In this regard, its degree of significance is representative in the area. Its bipartite division demonstrates a closer settlement and development of the land and, together with Samson's Cottage wall remains illustrates the development of Sydney at ever higher densities. In this regard, its degree of significance is representative in the area. The loss of the rear yards to public use has destroyed evidence of its former residential use. This increasing public pedestrianisation of formerly private areas both lessens the site's significance and increases the importance of maintaining what little of the site is left. In this regards, its degree of significance is representative in the area.Robertson & Hindmarsh 1998: 54-55 Samsons Cottage wall remains: 75.5 George Street can be considered to have high Historical Significance due to the unique archaeological assemblage recovered from the 1990 excavation.

However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither nor the complete bipartite graph as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions. As part of the Petersen family, plays a similar role as one of the forbidden minors for linkless embedding.. In other words, and as Conway and Gordon proved, every embedding of into three-dimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any three- dimensional embedding of contains a Hamiltonian cycle that is embedded in space as a nontrivial knot.

No matter which choice one makes of a color from the list of A and a color from the list of B, there will be some other vertex such that both of its choices are already used to color its neighbors. Thus, G is not 2-choosable. On the other hand, it is easy to see that G is 3-choosable: picking arbitrary colors for the vertices A and B leaves at least one available color for each of the remaining vertices, and these colors may be chosen arbitrarily. A list coloring instance on the complete bipartite graph K3,27 with three colors per vertex. No matter which colors are chosen for the three central vertices, one of the outer 27 vertices will be uncolorable, showing that the list chromatic number of K3,27 is at least four.

The first period can be defined as opposition to what is considered a move away of the Galician Nationalist Bloc from some of its key objectives, especially the National sovereignty and its leftist ideology. The political praxis of the BNG in the PSOE in the bipartite Xunta de Galicia (2005-2009) caused that militants of the Galician People's Union decided to leave that party to create a new organization within the BNG, under the name of Movemento pola Base (MpB). They were joined by independent militants of the BNG (not attached to any internal group or party) militants and members off the Confederación Intersindical Galega in the comarcas of Ferrolterra, Compostela and Vigo, among them Antolín Alcántara,Leader of the Confederación Intersindical Galega-Metal federation. Fermín Paz, Manuel Mera and Ramiro Oubiña, who were part of the executive of the CIG.

A toroidal embedding of K3,3 may be obtained by replacing the crossing by a tube, as described above, in which the two holes where the tube connects to the plane are placed along one of the crossing edges on either side of the crossing. Another way of changing the rules of the puzzle is to allow utility lines to pass through the cottages or utilities; this extra freedom allows the puzzle to be solved. Pál Turán's "brick factory problem" asks more generally for a formula for the minimum number of crossings in a drawing of the complete bipartite graph Ka,b in terms of the numbers of vertices a and b on the two sides of the bipartition. The utility graph K3,3 may be drawn with only one crossing, but not with zero crossings, so its crossing number is one..

An algorithm for listing all maximal independent sets or maximal cliques in a graph can be used as a subroutine for solving many NP- complete graph problems. Most obviously, the solutions to the maximum independent set problem, the maximum clique problem, and the minimum independent dominating problem must all be maximal independent sets or maximal cliques, and can be found by an algorithm that lists all maximal independent sets or maximal cliques and retains the ones with the largest or smallest size. Similarly, the minimum vertex cover can be found as the complement of one of the maximal independent sets. observed that listing maximal independent sets can also be used to find 3-colorings of graphs: a graph can be 3-colored if and only if the complement of one of its maximal independent sets is bipartite.

The Boundary Treaty of 1874 between Chile and Bolivia, also called the Treaty of Sucre, was signed in Sucre on August 6, 1874 by the Bolivian Minister of Foreign Affairs Mariano Baptista and the Chilean plenipotentiary minister Carlos Walker Martínez. It superseded the Boundary Treaty of 1866 between Chile and Bolivia and it kept the border between both countries at the 24° South parallel from the Pacific Ocean to the eastern border of Chile. The Treaty abolished the zone of bipartite tax collection on the export dues on minerals found between parallel 23°S and 25°S, and Bolivia promised explicitly in article 4 of the Treaty not to augment the existing taxes for twenty-five years on Chilean capital and industry. To realize its Saltpeter Monopoly, Peru tried fruitless to prevent the signing of the treaty.

Therefore, the planar graphs have a forbidden minor characterization, which in this case is given by Wagner's theorem: the set H of minor-minimal nonplanar graphs contains exactly two graphs, the complete graph K5 and the complete bipartite graph K3,3, and the planar graphs are exactly the graphs that do not have a minor in the set {K5, K3,3}. The existence of forbidden minor characterizations for all minor-closed graph families is an equivalent way of stating the Robertson–Seymour theorem. For, suppose that every minor-closed family F has a finite set H of minimal forbidden minors, and let S be any infinite set of graphs. Define F from S as the family of graphs that do not have a minor in S. Then F is minor-closed and has a finite set H of minimal forbidden minors.

In August 2011, Roger Colbeck and Renato Renner published a proof that any extension of quantum mechanical theory, whether using hidden variables or otherwise, cannot provide a more accurate prediction of outcomes, assuming that observers can freely choose the measurement settings. Colbeck and Renner write: "In the present work, we have ... excluded the possibility that any extension of quantum theory (not necessarily in the form of local hidden variables) can help predict the outcomes of any measurement on any quantum state. In this sense, we show the following: under the assumption that measurement settings can be chosen freely, quantum theory really is complete". In January 2013, Giancarlo Ghirardi and Raffaele Romano described a model which, "under a different free choice assumption [...] violates [the statement by Colbeck and Renner] for almost all states of a bipartite two-level system, in a possibly experimentally testable way".

The Boundary Treaty of 1866 between Chile and Bolivia, also called the Mutual Benefits Treaty, was signed in Santiago de Chile on August 10, 1866 by the Chilean Foreign Affairs Minister Alvaro Covarrubias and the Bolivian Plenipotentiary in Santiago Juan R. Muñoz Cabrera. It drew, for the first time, the border between both countries at the 24° South parallel from the Pacific Ocean to the eastern border of Chile and defined a zone of bipartite tax collection, the "Mutual Benefits zone", and tax preferences for articles from Bolivia and Chile. Despite increasing border tensions since the 1840s, both countries fought together against Spain in the Chincha Islands War (1864–65) and resolved the question under the Governments of Mariano Melgarejo in Bolivia and José Joaquín Pérez in Chile. But before long, both countries were discontented with it, and Peru and Bolivia signed a secret treaty against Chile in 1873.

RAC drawings of the complete graph K5 and the complete bipartite graph K3,4 In graph drawing, a RAC drawing of a graph is a drawing in which the vertices are represented as points, the edges are represented as straight line segments or polylines, at most two edges cross at any point, and when two edges cross they do so at right angles to each other. In the name of this drawing style, "RAC" stands for "right angle crossing". The right-angle crossing style and the name "RAC drawing" for this style were both formulated by ,. motivated by previous user studies showing that crossings with large angles are much less harmful to the readability of drawings than shallow crossings.. Even for planar graphs, allowing some right-angle crossings in a drawing of the graph can significantly improve measures of the drawing quality such as its area or angular resolution..

Monoamine oxidase B has a hydrophobic bipartite elongated cavity that (for the "open" conformation) occupies a combined volume close to 700 Å3. hMAO-A has a single cavity that exhibits a rounder shape and is larger in volume than the "substrate cavity" of hMAO-B. The first cavity of hMAO-B has been termed the entrance cavity (290 Å3), the second substrate cavity or active site cavity (~390 Å3) – between both an isoleucine199 side- chain serves as a gate. Depending on the substrate or bound inhibitor, it can exist in either an open or a closed form, which has been shown to be important in defining the inhibitor specificity of hMAO B. At the end of the substrate cavity is the FAD coenzyme with sites for favorable amine binding about the flavin involving two nearly parallel tyrosyl (398 and 435) residues that form what has been termed an aromatic cage.

Kullman, however, states that "Interestingly enough, Kuratowski did not publish a detailed proof that [ K3,3 is ] non-planar". One proof of the impossibility of finding a planar embedding of K3,3 uses a case analysis involving the Jordan curve theorem. In this solution, one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding. Alternatively, it is possible to show that any bridgeless bipartite planar graph with vertices and edges has by combining the Euler formula (where is the number of faces of a planar embedding) with the observation that the number of faces is at most half the number of edges (the vertices around each face must alternate between houses and utilities, so each face has at least four edges, and each edge belongs to exactly two faces).

Around that time, in 1991, Dov Dori, who then joined Technion – Israel Institute of Technology as faculty, realized that just as the procedural approach to software was not necessarily the most adequate one, neither was the "pure" OO approach, which puts objects as the sole "first class" citizens, with "methods" (or "services", or "operations") being their second-class subordinate procedures. When he and colleagues from University of Washington were trying to model a system for automated transformation of engineering drawings to CAD models, he realized that not all the boxes in their model were really objects; some were things that happen to objects. When he circled those things, a bipartite graph emerged, in which the nodes representing objects—the things that exist—were mediated by those circled nodes, which were identified as processes—the things that transform the objects. Dori published the first paper on OPM in 1995.

By similar reasoning, the complete bipartite graph has no 2-basis: is four-dimensional, and each nontrivial vector in has nonzero coordinates for at least four edges, so any augmented basis would have at least 20 nonzeros, exceeding the 18 nonzeros that would be allowed if each of the nine edges were nonzero in at most two basis vectors. Since the property of having a 2-basis is minor-closed and is not true of the two minor-minimal nonplanar graphs and , it is also not true of any other nonplanar graph. provided another proof, based on algebraic topology. He uses a slightly different formulation of the planarity criterion, according to which a graph is planar if and only if it has a set of (not necessarily simple) cycles covering every edge exactly twice, such that the only nontrivial relation among these cycles in is that their sum be zero.

A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron) In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.. Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.

In 2012, he published a bipartite book about the histories of Kazakhstan and Kyrgyzstan. His latest publication before this work was “Cold War II: cries in the desert - or how to counterbalance NATO’s propaganda from Ukraine to Central Asia”, published by Hertfordshire Press, England. As an explorative reporter and commentator, Charles van der Leeuw has consistently stressed the need to obtain insight in “the other side of the story” i.e. facts, trends and views opposed to what mainstream media usually offer their audiences. This was already reflected in “Koeweit brandt” in which he put question marks behind the international outcry against Saddam Hussein when he invaded the tiny emirate and the cheers in favour of America’s crackdown in Iraq that followed. In the case of Nagorno-Karabakh, he put the spotlight on the Azeri side of the story, in an attempt to counterbalance the powerful global Armenian lobby depicting the conflict as “Turkish” aggression against the innocent Armenians.

A perfect graph is a graph in which, for every induced subgraph, the size of the maximum clique equals the minimum number of colors in a coloring of the graph; perfect graphs include many well-known graph classes including the bipartite graphs, chordal graphs, and comparability graphs. In his 1961 and 1963 works defining for the first time this class of graphs, Claude Berge observed that it is impossible for a perfect graph to contain an odd hole, an induced subgraph in the form of an odd-length cycle graph of length five or more, because odd holes have clique number two and chromatic number three. Similarly, he observed that perfect graphs cannot contain odd antiholes, induced subgraphs complementary to odd holes: an odd antihole with 2k + 1 vertices has clique number k and chromatic number k + 1, which is again impossible for perfect graphs. The graphs having neither odd holes nor odd antiholes became known as the Berge graphs.

In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K5 and the complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of one of these two graphs as a subgraph (in which case it does not belong to the planar graphs).

The smallest 3-regular matchstick graph is formed from two copies of the diamond graph placed in such a way that corresponding vertices are at unit distance from each other; its bipartite double cover is the 8-crossed prism graph. In 1986, Heiko Harborth presented the graph that would bear his name, the Harborth Graph. With 104 edges and 52 vertices, is the smallest known example of a 4-regular matchstick graph.. As cited in: It is a rigid graph.. For additional details see Gerbracht's earlier preprint "Minimal Polynomials for the Coordinates of the Harborth Graph" (2006), arXiv:math/0609360. Every 4-regular matchstick graph contains at least 20 vertices.. Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61 and 62. The graphs with 54, 57, 65, 67, 73, 74, 77 and 85 vertices were first published in 2016.

Origins in bacteria are either continuous or bipartite and contain three functional elements that control origin activity: conserved DNA repeats that are specifically recognized by DnaA (called DnaA-boxes), an AT-rich DNA unwinding element (DUE), and binding sites for proteins that help regulate replication initiation. Interactions of DnaA both with the double-stranded (ds) DnaA-box regions and with single- stranded (ss) DNA in the DUE are important for origin activation and are mediated by different domains in the initiator protein: a Helix-turn-helix (HTH) DNA binding element and an ATPase associated with various cellular activities (AAA+) domain, respectively. While the sequence, number, and arrangement of origin-associated DnaA-boxes vary throughout the bacterial kingdom, their specific positioning and spacing in a given species are critical for oriC function and for productive initiation complex formation. Among bacteria, E. coli is a particularly powerful model system to study the organization, recognition, and activation mechanism of replication origins.

The nine-vertex Paley graph, a balanced tripartite graph with 18 edges, each belonging to exactly one triangle Several views of the Brouwer–Haemers graph, a non-tripartite 20-regular graph with 81 vertices in which each edge belongs to exactly one triangle In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle. Equivalently it asks for the maximum number of edges in a balanced bipartite graph whose edges can be partitioned into a linear number of induced matchings, or the maximum number of triples one can choose from n points so that every six points contain at most two triples. The problem is named after Imre Z. Ruzsa and Endre Szemerédi, who first proved that its answer is smaller than n^2 by a slowly-growing (but still unknown) factor.

The Desargues graph is a symmetric graph: it has symmetries that take any vertex to any other vertex and any edge to any other edge. Its symmetry group has order 240, and is isomorphic to the product of a symmetric group on 5 points with a group of order 2\. One can interpret this product representation of the symmetry group in terms of the constructions of the Desargues graph: the symmetric group on five points is the symmetry group of the Desargues configuration, and the order-2 subgroup swaps the roles of the vertices that represent points of the Desargues configuration and the vertices that represent lines. Alternatively, in terms of the bipartite Kneser graph, the symmetric group on five points acts separately on the two-element and three-element subsets of the five points, and complementation of subsets forms a group of order two that transforms one type of subset into the other.

In the case of the ménage problem, the vertices of the graph represent men and women, and the edges represent pairs of men and women who are allowed to sit next to each other. This graph is formed by removing the perfect matching formed by the male-female couples from a complete bipartite graph that connects every man to every woman. Any valid seating arrangement can be described by the sequence of people in order around the table, which forms a Hamiltonian cycle in the graph. However, two Hamiltonian cycles are considered to be equivalent if they connect the same vertices in the same cyclic order regardless of the starting vertex, while in the ménage problem the starting position is considered significant: if, as in Alice's tea party, all the guests shift their positions by one seat, it is considered a different seating arrangement even though it is described by the same cycle.

Not all graph families have local structures. For some families, a simple counting argument proves that adjacency labeling schemes do not exist: only bits may be used to represent an entire graph, so a representation of this type can only exist when the number of -vertex graphs in the given family is at most . Graph families that have larger numbers of graphs than this, such as the bipartite graphs or the triangle-free graphs, do not have adjacency labeling schemes.. However, even families of graphs in which the number of graphs in the family is small might not have an adjacency labeling scheme; for instance, the family of graphs with fewer edges than vertices has -vertex graphs but does not have an adjacency labeling scheme, because one could transform any given graph into a larger graph in this family by adding a new isolated vertex for each edge, without changing its labelability. Kannan et al.

Let G be a graph and let T be a Trémaux tree of G. The graph G is planar if and only if there exists a partition of the cotree edges of G into two classes so that any two edges belong to a same class if they are T-alike and any two edges belong to different classes if they are T-opposite. This characterization immediately leads to an (inefficient) planarity test: determine for all pairs of edges whether they are T-alike or T-opposite, form an auxiliary graph that has a vertex for each connected component of T-alike edges and an edge for each pair of T-opposite edges, and check whether this auxiliary graph is bipartite. Making this algorithm efficient involves finding a subset of the T-alike and T-opposite pairs that is sufficient to carry out this method without determining the relation between all edge pairs in the input graph.

Wagner's theorem states that every graph has either a planar embedding, or a minor of one of two types, the complete graph K5 or the complete bipartite graph K3,3. (It is also possible for a single graph to have both types of minor.) If a given graph is planar, so are all its minors: vertex and edge deletion obviously preserve planarity, and edge contraction can also be done in a planarity-preserving way, by leaving one of the two endpoints of the contracted edge in place and routing all of the edges that were incident to the other endpoint along the path of the contracted edge. A minor-minimal non-planar graph is a graph that is not planar, but in which all proper minors (minors formed by at least one deletion or contraction) are planar. Another way of stating Wagner's theorem is that there are only two minor-minimal non-planar graphs, K5 and K3,3.

Brouwer has confirmed by computation that the conjecture is valid for all graphs with at most 10 vertices. It is also known that the conjecture is valid for any number of vertices if t = 1, 2, n − 1, and n. For certain types of graphs, Brouwer's conjecture is known to be valid for all t and for any number of vertices. In particular, it is known that is valid for trees, and for unicyclic and bicyclic graphs. It was also proved that Brouwer’s conjecture holds for two large families of graphs; the first family of graphs is obtained from a clique by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph.

"Traditional" Balto-Slavic tree model This bipartite division into Baltic and Slavic was first challenged in the 1960s, when Vladimir Toporov and Vyacheslav Ivanov observed that the apparent difference between the "structural models" of the Baltic languages and the Slavic languages is the result of the innovative nature of Proto-Slavic, and that the latter had evolved from an earlier stage which conformed to the more archaic "structural model" of the Proto-Baltic dialect continuum.Бирнбаум Х. О двух направлениях в языковом развитии // Вопросы языкознания, 1985, № 2, стр. 36 Frederik Kortlandt (1977, 2018) has proposed that West Baltic and East Baltic are in fact not more closely related to each other than either of them is related to Slavic, and Balto-Slavic therefore can be split into three equidistant branches: East Baltic, West Baltic and Slavic. Alternative Balto-Slavic tree model Although supported by a number of scholars, Kordtlandt's hypothesis is still a minority view.

System of the Moroccan Walls in Western Sahara set up in the 1980s Commemoration of the 30th independence day from Spain in the Liberated Territories (2005) In the waning days of General Franco's rule, and after the Green March, the Spanish government signed a tripartite agreement with Morocco and Mauritania as it moved to transfer the territory on 14 November 1975. The accords were based on a bipartite administration, and Morocco and Mauritania each moved to annex the territories, with Morocco taking control of the northern two-thirds of Western Sahara as its Southern Provinces, and Mauritania taking control of the southern third as Tiris al-Gharbiyya. Spain terminated its presence in Spanish Sahara within three months, repatriating Spanish remains from its cemeteries.Tomás Bárbulo, "La historia prohibida del Sáhara Español," Destino, Imago mundi, Volume 21, 2002, Page 292 The Moroccan and Mauritanian annexations were resisted by the Polisario Front, which had gained backing from Algeria.

In general, a graph may have multiple double covers that are different from the bipartite double cover.. In the following figure, the graph C is a double cover of the graph H: # The graph C is a covering graph of H: there is a surjective local isomorphism f from C to H, the one indicated by the colours. For example, f maps both blue nodes in C to the blue node in H. Furthermore, let X be the neighbourhood of a blue node in C and let Y be the neighbourhood of the blue node in H; then the restriction of f to X is a bijection from X to Y. In particular, the degree of each blue node is the same. The same applies to each colour. # The graph C is a double cover (or 2-fold cover or 2-lift) of H: the preimage of each node in H has size 2.

The butterfly graph (left) and diamond graph (right), forbidden minors for pseudoforests Forming a minor of a pseudoforest by contracting some of its edges and deleting others produces another pseudoforest. Therefore, the family of pseudoforests is closed under minors, and the Robertson–Seymour theorem implies that pseudoforests can be characterized in terms of a finite set of forbidden minors, analogously to Wagner's theorem characterizing the planar graphs as the graphs having neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. As discussed above, any non-pseudoforest graph contains as a subgraph a handcuff, figure 8, or theta graph; any handcuff or figure 8 graph may be contracted to form a butterfly graph (five-vertex figure 8), and any theta graph may be contracted to form a diamond graph (four-vertex theta graph),For this terminology, see the list of small graphs from the Information System on Graph Class Inclusions. However, butterfly graph may also refer to a different family of graphs related to hypercubes, and the five-vertex figure 8 is sometimes instead called a bowtie graph.

Hintikka's proposal was met with skepticism by a number of logicians because some first-order sentences like the one below appear to capture well enough the natural language Hintikka sentence. : [\forall x_1 \, \exists y_1 \, \forall x_2 \, \exists y_2\, \varphi (x_1, x_2, y_1, y_2)] \wedge [\forall x_2 \, \exists y_2 \, \forall x_1 \, \exists y_1\, \varphi (x_1, x_2, y_1, y_2)] where : \varphi (x_1, x_2, y_1, y_2) denotes : (V(x_1) \wedge T(x_2)) \rightarrow (R(x_1,y_1) \wedge R(x_2,y_2) \wedge H(y_1, y_2) \wedge H(y_2, y_1)) Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trained in logic found that they are more likely to assign models matching the "bidirectional" first-order sentence rather than branching-quantifier sentence to several natural-language constructs derived from the Hintikka sentence. For instance students were shown undirected bipartite graphs--with squares and circles as vertices--and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were correctly describing the diagrams.

Robertson's irreducible apex graph, showing that the YΔY- reducible graphs have additional forbidden minors beyond those in the Petersen family. A minor of a graph G is another graph formed from G by contracting and removing edges. As the Robertson–Seymour theorem shows, many important families of graphs can be characterized by a finite set of forbidden minors: for instance, according to Wagner's theorem, the planar graphs are exactly the graphs that have neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. Neil Robertson, Paul Seymour, and Robin Thomas used the Petersen family as part of a similar characterization of linkless embeddings of graphs, embeddings of a given graph into Euclidean space in such a way that every cycle in the graph is the boundary of a disk that is not crossed by any other part of the graph.. Horst Sachs had previously studied such embeddings, shown that the seven graphs of the Petersen family do not have such embeddings, and posed the question of characterizing the linklessly embeddable graphs by forbidden subgraphs.. Robertson et al.

Because it is the bipartite half of a distance- regular graph, the halved cube graph is itself distance-regular.. And because it contains a hypercube as a spanning subgraph, it inherits from the hypercube all monotone graph properties, such as the property of containing a Hamiltonian cycle. As with the hypercube graphs, and their isometric (distance-preserving) subgraphs the partial cubes, a halved cube graph may be embedded isometrically into a real vector space with the Manhattan metric (L1 distance function). The same is true for the isometric subgraphs of halved cube graphs, which may be recognized in polynomial time; this forms a key subroutine for an algorithm which tests whether a given graph may be embedded isometrically into a Manhattan metric.. For every halved cube graph of order five or more, it is possible to (improperly) color the vertices with two colors, in such a way that the resulting colored graph has no nontrivial symmetries. For the graphs of order three and four, four colors are needed to eliminate all symmetries..

Graham and Pollak study a more general graph labeling problem, in which the vertices of a graph should be labeled with equal-length strings of the characters "0", "1", and "✶", in such a way that the distance between any two vertices equals the number of string positions where one vertex is labeled with a 0 and the other is labeled with a 1. A labeling like this with no "✶" characters would give an isometric embedding into a hypercube, something that is only possible for graphs that are partial cubes, and in one of their papers Graham and Pollak call a labeling that allows "✶" characters an embedding into a "squashed cube". For each position of the label strings, one can define a complete bipartite graph in which one side of the bipartition consists of the vertices labeled with 0 in that position and the other side consists of the vertices labeled with 1, omitting the vertices labeled "✶". For the complete graph, every two vertices are at distance one from each other, so every edge must belong to exactly one of these complete graphs.

A partition of the complete bipartite graph K4,4 into three forests, showing that it has arboricity three. The Thue number of a graph is the number of colors required in an edge coloring meeting the stronger requirement that, in every even-length path, the first and second halves of the path form different sequences of colors. The arboricity of a graph is the minimum number of colors required so that the edges of each color have no cycles (rather than, in the standard edge coloring problem, having no adjacent pairs of edges). That is, it is the minimum number of forests into which the edges of the graph may be partitioned into.. Unlike the chromatic index, the arboricity of a graph may be computed in polynomial time.. List edge-coloring is a problem in which one is given a graph in which each edge is associated with a list of colors, and must find a proper edge coloring in which the color of each edge is drawn from that edge's list.

Tait's motivation for studying the ménage problem came from trying to find a complete listing of mathematical knots with a given number of crossings, say n. In Dowker notation for knot diagrams, an early form of which was used by Tait, the 2n points where a knot crosses itself, in consecutive order along the knot, are labeled with the 2n numbers from 1 to 2n. In a reduced diagram, the two labels at a crossing cannot be consecutive, so the set of pairs of labels at each crossing, used in Dowker notation to represent the knot, can be interpreted as a perfect matching in a graph that has a vertex for every number in the range from 1 to 2n and an edge between every pair of numbers that has different parity and are non-consecutive modulo 2n. This graph is formed by removing a Hamiltonian cycle (connecting consecutive numbers) from a complete bipartite graph (connecting all pairs of numbers with different parity), and so it has a number of matchings equal to a ménage number.

An (N,M,D,K,\epsilon) -extractor is a bipartite graph with N nodes on the left and M nodes on the right such that each node on the left has D neighbors (on the right), which has the added property that for any subset A of the left vertices of size at least K, the distribution on right vertices obtained by choosing a random node in A and then following a random edge to get a node x on the right side is \epsilon-close to the uniform distribution in terms of total variation distance. A disperser is a related graph. An equivalent way to view an extractor is as a bivariate function :E : [N] \times [D] \rightarrow [M] in the natural way. With this view it turns out that the extractor property is equivalent to: for any source of randomness X that gives n bits with min-entropy \log K, the distribution E(X,U_D) is \epsilon-close to U_M, where U_T denotes the uniform distribution on [T].

When the non-zero elements of the measurement matrix are chosen randomly from a continuous distribution, then it can be shown that if one variable node receives equal messages divided by the edge weights from its neighbors then this variable node is the only unique variable connected to all of those check nodes, therefore, the rule can be applied using a local decision approach, and the variable node can verify itself without further knowledge about the other connections of those check nodes. Moreover, the second part of the ECN rule is not necessary to be implemented as the non-zero verified variable node in the ECN rule will be removed from the bipartite graph in the next iteration and ZCN rule will be enough to verify all the zero valued variable nodes remained from those equations with the same right hand side. All in all, when the non- zero elements of the measurement matrix are chosen form a continuous distribution then the SBB and XH algorithm that use ECN rule can be implemented efficiently.

A 2-dimensional matching can be defined in a completely analogous manner. Let X and Y be finite, disjoint sets, and let T be a subset of X × Y. Now M ⊆ T is a 2-dimensional matching if the following holds: for any two distinct pairs (x1, y1) ∈ M and (x2, y2) ∈ M, we have x1 ≠ x2 and y1 ≠ y2. In the case of 2-dimensional matching, the set T can be interpreted as the set of edges in a bipartite graph G = (X, Y, T); each edge in T connects a vertex in X to a vertex in Y. A 2-dimensional matching is then a matching in the graph G, that is, a set of pairwise non-adjacent edges. Hence 3-dimensional matchings can be interpreted as a generalization of matchings to hypergraphs: the sets X, Y, and Z contain the vertices, each element of T is a hyperedge, and the set M consists of pairwise non-adjacent edges (edges that do not have a common vertex).

In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem.) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering.. Equivalently, every family of graphs that is closed under minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K5 or the complete bipartite graph K3,3 as minors. The Robertson–Seymour theorem is named after mathematicians Neil Robertson and Paul D. Seymour, who proved it in a series of twenty papers spanning over 500 pages from 1983 to 2004.; . Before its proof, the statement of the theorem was known as Wagner's conjecture after the German mathematician Klaus Wagner, although Wagner said he never conjectured it.. A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 1960 independently by Joseph Kruskal and S. Tarkowski.

In his 1881 parallel work, The Ancient Bronze Implements, he affirmed and further defined the three periods, strangely enough recusing himself from his previous terminology, Early, Middle and Late Bronze Age (the current forms) in favor of "an earlier and later stage" and "middle". He uses Bronze Age, Bronze Period, Bronze-using Period and Bronze Civilization interchangeably. Apparently Evans was sensitive of what had gone before, retaining the terminology of the bipartite system while proposing a tripartite one. After stating a catalogue of types of bronze implements he defines his system: > The Bronze Age of Britain may, therefore, be regarded as an aggregate of > three stages: the first, that characterized by the flat or slightly flanged > celts, and the knife-daggers ... the second, that characterized by the more > heavy dagger-blades and the flanged celts and tanged spear-heads or daggers, > ... and the third, by palstaves and socketed celts and the many forms of > tools and weapons, ... It is in this third stage that the bronze sword and > the true socketed spear-head first make their advent.

If a family F of graphs is closed under taking minors (every minor of a member of F is also in F), then by the Robertson–Seymour theorem F can be characterized as the graphs that do not have any minor in X, where X is a finite set of forbidden minors.. For instance, Wagner's theorem states that the planar graphs are the graphs that have neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. In many cases, the properties of F and the properties of X are closely related, and the first such result of this type was by , and relates bounded pathwidth with the existence of a forest in the family of forbidden minors. Specifically, define a family F of graphs to have bounded pathwidth if there exists a constant p such that every graph in F has pathwidth at most p. Then, a minor-closed family F has bounded pathwidth if and only if the set X of forbidden minors for F includes at least one forest.

A common way to form covering graphs uses voltage graphs, in which the darts of the given graph G (that is, pairs of directed edges corresponding to the undirected edges of G) are labeled with inverse pairs of elements from some group. The derived graph of the voltage graph has as its vertices the pairs (v,x) where v is a vertex of G and x is a group element; a dart from v to w labeled with the group element y in G corresponds to an edge from (v,x) to (w,xy) in the derived graph. The universal cover can be seen in this way as a derived graph of a voltage graph in which the edges of a spanning tree of the graph are labeled by the identity element of the group, and each remaining pair of darts is labeled by a distinct generating element of a free group. The bipartite double can be seen in this way as a derived graph of a voltage graph in which each dart is labeled by the nonzero element of the group of order two.

These verses, it is argued, indicate that death is only a period or form of slumber. Adventists teach that the resurrection of the righteous will take place shortly after the second coming of Jesus, as described in Revelation 20:4–6 that follows Revelation 19:11–16, whereas the resurrection of the wicked will occur after the millennium, as described in Revelation 20:5 and 20:12–13 that follow Revelation 20:4 and 6–7, though Revelation 20:12–13 and 15 actually describe a mixture of saved and condemned people being raised from the dead and judged. Adventists reject the traditional doctrine of hell as a state of everlasting conscious torment, believing instead that the wicked will be permanently destroyed after the millennium by the lake of fire, which is called 'the second death' in Revelation 20:14. Those Adventist doctrines about death and hell reflect an underlying belief in: (a) conditional immortality (or conditionalism), as opposed to the immortality of the soul; and (b) the monistic nature of human beings, in which the soul is not separable from the body, as opposed to bipartite or tripartite conceptions, in which the soul is separable.

Any semisimple algebra over the complex numbers C of finite dimension can be expressed as a direct sum ⊕k Mnk(C) of matrix algebras, and the C-algebra homomorphisms between two such algebras up to inner automorphisms on both sides are completely determined by the multiplicity number between 'matrix algebra' components. Thus, an injective homomorphism of ⊕k=1i Mnk(C) into ⊕l=1j Mml(C) may be represented by a collection of positive numbers ak, l satisfying ∑ nk ak, l ≤ ml. (The equality holds if and only if the homomorphism is unital; we can allow non-injective homomorphisms by allowing some ak,l to be zero.) This can be illustrated as a bipartite graph having the vertices marked by numbers (nk)k on one hand and the ones marked by (ml)l on the other hand, and having ak, l edges between the vertex nk and the vertex ml. Thus, when we have a sequence of finite- dimensional semisimple algebras An and injective homomorphisms φn : An' → An+1: between them, we obtain a Bratteli diagram by putting : Vn = the set of simple components of An (each isomorphic to a matrix algebra), marked by the size of matrices.

The "girth" terminology generalizes the use of girth in graph theory, meaning the length of the shortest cycle in a graph: the girth of a graphic matroid is the same as the girth of its underlying graph.. The girth of other classes of matroids also corresponds to important combinatorial problems. For instance, the girth of a co-graphic matroid (or the cogirth of a graphic matroid) equals the edge connectivity of the underlying graph, the number of edges in a minimum cut of the graph. The girth of a transversal matroid gives the cardinality of a minimum Hall set in a bipartite graph: this is a set of vertices on one side of the bipartition that does not form the set of endpoints of a matching in the graph.. Any set of points in Euclidean space gives rise to a real linear matroid by interpreting the Cartesian coordinates of the points as the vectors of a matroid representation. The girth of the resulting matroid equals one plus the dimension of the space when the underlying set of point is in general position, and is smaller otherwise.