300 Sentences With "directed graph" | Random Sentence Generator

DIRECTED GRAPH In discrete math, a directed graph shows the relationship between pairs of objects.

This directed graph quite easily shows us which continents have the most and least migration.

In a directed graph (or digraph), the directed line segments are "one-sided" like one-way streets.

If the links between nodes are bidirectional, we say that the graph is undirected; otherwise, it's a directed graph.

One possibility here is to use directed graph structures, which represent all the genetic variation of a population in a lightweight data format.

This week's graph is a directed graph (see Stat Nuggets for full explanation) of this surge in passenger demand during this one week.

In the Thanksgiving passenger graph, the directed graph shows the increase in traffic for Thanksgiving between flight origins and destinations, which are the vertices.

A symmetric directed graph (that is, a directed graph in which the reverse of every edge is also an edge) is sometimes also called a "bidirected graph".

The degree sequence of a directed graph is the list of its indegree and outdegree pairs; for the above example we have degree sequence ((2, 0), (2, 2), (0, 2), (1, 1)). The degree sequence is a directed graph invariant so isomorphic directed graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a directed graph; in some cases, non-isomorphic digraphs have the same degree sequence. The directed graph realization problem is the problem of finding a directed graph with the degree sequence a given sequence of positive integer pairs.

The adjacency matrix of a directed graph is unique up to identical permutation of rows and columns. Another matrix representation for a directed graph is its incidence matrix. See direction for more definitions.

An accessible pointed graph is a directed graph with a distinguished vertex (the "root") such that for any node in the graph there is at least one path in the directed graph from the root to that node. The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of a unique set. For example, the directed graph with only one node and an edge from that node to itself corresponds to a set of the form x = {x}.

The definition of a composite service contains an implicit directed graph of inner service dependencies. The runtime environment for SOP can create an execution graph based on this directed graph by automatically instantiating and running inner services in parallel whenever possible.

Directed Graph Markup Language (DGML) is an XML-based file format for directed graphs.

One definition of an oriented graph is that it is a directed graph in which at most one of and may be edges of the graph. That is, it is a directed graph that can be formed as an orientation of an undirected (simple) graph. Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a given undirected graph or multigraph.

A directed graph with three vertices (blue circles) and three edges (black arrows). In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also called links or lines), and for a directed graph are also known as arrows.

For a directed graph, a loop adds one to the in degree and one to the out degree.

There are two algorithms available for directed graph: Camerini's algorithm for finding MBSA and another from Gabow and Tarjan.

A simple directed graph. Here the double-headed arrow represents two distinct edges, one for each direction. In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.

Versions of these definitions are also used for directed graphs. Like an undirected graph, a directed graph consists of vertices and edges, but each edge is directed from one of its endpoints to the other endpoint. A directed pseudoforest is a directed graph in which each vertex has at most one outgoing edge; that is, it has outdegree at most one. A directed 1-forest - most commonly called a functional graph (see below), sometimes maximal directed pseudoforest - is a directed graph in which each vertex has outdegree exactly one.

A directed graph without directed cycles is called a directed acyclic graph. A connected graph without cycles is called a tree.

In mathematics, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges as possible, such that if there is a (directed) path from vertex v to vertex w in D, then there is also such a path in the reduction. Transitive reductions were introduced by , who provided tight bounds on the computational complexity of constructing them. More technically, the reduction is a directed graph that has the same reachability relation as D. Equivalently, D and its transitive reduction should have the same transitive closure as each other, and its transitive reduction should have as few edges as possible among all graphs with this property. The transitive reduction of a finite directed acyclic graph (a directed graph without directed cycles) is unique and is a subgraph of the given graph.

A Hamiltonian cycle in a directed graph is a cycle that passes through each vertex of the graph exactly once. The following Lparse program can be used to find a Hamiltonian cycle in a given directed graph if it exists; we assume that 0 is one of the vertices. {in(X,Y)} :- e(X,Y). :- 2 {in(X,Y) : e(X,Y)}, v(X).

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.

A directed graph (blue and black) and its condensation (yellow). The strongly connected components (subsets of blue vertices within each yellow vertex) form the blocks of a partition giving rise to the quotient. The condensation of a directed graph is the quotient graph where the strongly connected components form the blocks of the partition. This construction can be used to derive a directed acyclic graph from any directed graph.. The result of one or more edge contractions in an undirected graph G is a quotient of G, in which the blocks are the connected components of the subgraph of G formed by the contracted edges.

Transitive reduction can be defined for an abstract binary relation on a set, by interpreting the pairs of the relation as arcs in a directed graph.

The above definitions can also be applied to directed graphs. An ear would then be a directed path where all internal vertices have indegree and outdegree equal to 1. A directed graph is strongly connected if it contains a directed path from every vertex to every other vertex. Then we have the following theorem : :A directed graph is strongly connected if and only if it has an ear decomposition.

A closure of a directed graph is a set of vertices with no outgoing edges. That is, the graph should have no edges that start within the closure and end outside the closure. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved in polynomial time using a reduction to the maximum flow problem.

The following theorems can be regarded as directed versions: :Ghouila-Houiri (1960). A strongly connected simple directed graph with n vertices is Hamiltonian if every vertex has a full degree greater than or equal to n. :Meyniel (1973). A strongly connected simple directed graph with n vertices is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to .

The auction algorithm of Bertsekas for finding shortest paths within a directed graph is reputed to perform very well on random graphs and on problems with few destinations.

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.

A directed graph is weakly connected (or just connected p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y and a directed path from y to x for every pair of vertices }.

The webgraph describes the directed links between pages of the World Wide Web. A graph, in general, consists of several vertices, some pairs connected by edges. In a directed graph, edges are directed lines or arcs. The webgraph is a directed graph, whose vertices correspond to the pages of the WWW, and a directed edge connects page X to page Y if there exists a hyperlink on page X, referring to page Y.

A data flow graph is a labeled directed graph. Each node is labeled by a type indicating its functionality, and each edge is labeled by a number indicating its delay.

The conjecture was disproved by Aanderaa, who exhibited a directed graph property (the property of containing a "sink") which required only O(n) queries to test. A sink, in a directed graph, is a vertex of indegree n-1 and outdegree 0. This property can be tested with less than 3n queries . An undirected graph property which can also be tested with O(n) queries is the property of being a scorpion graph, first described in .

In a directed graph, an ordered pair of vertices is called strongly connected if a directed path leads from x to y. Otherwise, the ordered pair is called weakly connected if an undirected path leads from x to y after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called disconnected. A strongly connected graph is a directed graph in which every ordered pair of vertices in the graph is strongly connected.

However, the same definitions apply to directed graphs and a directed graph is also equivalent to a unique core. Every graph and every directed graph contains its core as a retract and as an induced subgraph. For example, all complete graphs Kn and all odd cycles (cycle graphs of odd length) are cores. Every 3-colorable graph G that contains a triangle (that is, has the complete graph K3 as a subgraph) is homomorphically equivalent to K3.

According to the solution of this problem , a strongly connected directed graph in which all vertices have the same outdegree has a synchronizable edge coloring if and only if it is aperiodic.

Using a directed graph, the probabilities of the possible states a hypothetical stock market can exhibit is represented. The matrix on the left shows how probabilities corresponding to different states can be arranged in matrix form. A state diagram for a simple example is shown in the figure on the right, using a directed graph to picture the state transitions. The states represent whether a hypothetical stock market is exhibiting a bull market, bear market, or stagnant market trend during a given week.

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..

In computer science, st-connectivity or STCON is a decision problem asking, for vertices s and t in a directed graph, if t is reachable from s. Formally, the decision problem is given by :.

The additional constraint of the acycilicity of the directed graph is known as dag realization. proved the NP-completeness of this problem. showed that the class of opposed sequences is in P. The problem uniform sampling a directed graph to a fixed degree sequence is to construct a solution for the digraph realization problem with the additional constraint that such each solution comes with the same probability. This problem was shown to be in FPTAS for regular sequences by The general problem is still unsolved.

A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, each edge has an orientation, from one vertex to another vertex. A path in a directed graph is a sequence of edges having the property that the ending vertex of each edge in the sequence is the same as the starting vertex of the next edge in the sequence; a path forms a cycle if the starting vertex of its first edge equals the ending vertex of its last edge. A directed acyclic graph is a directed graph that has no cycles.... Adding the red edges to the blue directed acyclic graph produces another DAG, the transitive closure of the blue graph.

A polytree is a directed graph formed by orienting the edges of a free tree.. Every polytree is a DAG. In particular, this is true of the arborescences formed by directing all edges outwards from the roots of a tree. A multitree (also called a strongly unambiguous graph or a mangrove) is a directed graph in which there is at most one directed path (in either direction) between any two vertices; equivalently, it is a DAG in which, for every vertex , the subgraph reachable from forms a tree..

To visualize the game, a directed graph can be constructed whose nodes are each cities of the world. An arrow is added from node N1 to node N2 if and only if the city labeling N2 starts with the letter that ending the name of the city labeling node N1. In other words, we draw an arrow from one city to another if the first can lead to the second according to the game rules. Each alternate edge in the directed graph corresponds to each player (for a two player game).

A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge with any one of the two possible orientations. Many of the important properties of tournaments were first investigated by in order to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things.

Martin Tompa proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12. Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p.

In graph theory, an arborescence is a directed graph in which, for a vertex u called the root and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph. Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist. Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.

This directed graph has no cycles: it is not possible to get from any vertex (point) back to that same point, following the connections in the direction indicated by the arrows. A feedback arc set (FAS) or feedback edge set is a set of edges which, when removed from the graph, leaves an acyclic graph. Put another way, it is a set containing at least one edge of every cycle in the graph. In graph theory, a directed graph may contain directed cycles, a closed one-way path of edges.

An oriented graph is a finite directed graph obtained from a simple undirected graph by assigning an orientation to each edge. Equivalently, it is a directed graph that has no self-loops, no parallel edges, and no two-edge cycles. The first neighborhood of a vertex v (also called its open neighborhood) consists of all vertices at distance one from v, and the second neighborhood of v consists of all vertices at distance two from v. These two neighborhoods form disjoint sets, neither of which contains v itself.

Cederbaum's theorem applies to a particular type of directed graph: . V is the set of nodes. E is the a set of directed edges: E = (a, b) \in V \times V . A positive weight is associated with each edge: .

NB: Wright's rules assume a model without feedback loops: the directed graph of the model must contain no cycles, i.e. it is a directed acyclic graph, which has been extensively studied in the causal analysis framework of Judea Pearl.

A vector addition system with states is a VAS equipped with control states. More precisely, it is a finite directed graph with arcs labelled by integer vectors. VASS have the same restriction that the counter values should never drop below zero.

In mathematics, and particularly in category theory, a polygraph is a generalisation of a directed graph. It is also known as a computad. They were introduced as "polygraphs" by Albert BurroniA. Burroni. Higher-dimensional word problems with applications to equational logic.

Microsoft Cognitive Toolkit, previously known as CNTK and sometimes styled as The Microsoft Cognitive Toolkit, is a deprecated deep learning framework developed by Microsoft Research. Microsoft Cognitive Toolkit describes neural networks as a series of computational steps via a directed graph.

In graph theory, the cycle rank of a directed graph is a digraph connectivity measure proposed first by Eggan and Büchi . Intuitively, this concept measures how close a digraph is to a directed acyclic graph (DAG), in the sense that a DAG has cycle rank zero, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. The cycle rank of a directed graph is closely related to the tree-depth of an undirected graph and to the star height of a regular language. It has also found use in sparse matrix computations (see ) and logic .

Euler tour of a tree, with edges labeled to show the order in which they are traversed by the tour The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree. The ETT allows for efficient, parallel computation of solutions to common problems in algorithmic graph theory.

In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow.

Equivalently, a strongly connected component of a directed graph G is a subgraph that is strongly connected, and is maximal with this property: no additional edges or vertices from G can be included in the subgraph without breaking its property of being strongly connected. The collection of strongly connected components forms a partition of the set of vertices of G. The yellow directed acyclic graph is the condensation of the blue directed graph. It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex. If each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G. A directed graph is acyclic if and only if it has no strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every nontrivial strongly connected component contains at least one directed cycle.

In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced in , and has seen many uses in the field of universal algebra since then.

The digraph realization problem is a decision problem in graph theory. Given pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)), the problem asks whether there is a labeled simple directed graph such that each vertex v_i has indegree a_i and outdegree b_i.

CPAchecker operates on a control-flow automata (CFA); before a given C program can be analysed by the CPA algorithm, it gets transformed into a CFA. A CFA is a directed graph whose edges represent either assumptions or assignments and its nodes represent program locations.

An object graph is a directed graph, which might be cyclic. When stored in RAM, objects occupy different segments of the memory with their attributes and function table, while relationships are represented by pointers or a different type of global handler in higher-level languages.

A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components.

A directed graph with a synchronizing coloring The image to the right shows a directed graph on eight vertices in which each vertex has out-degree 2\. (Each vertex in this case also has in-degree 2, but that is not necessary for a synchronizing coloring to exist.) The edges of this graph have been colored red and blue to create a synchronizing coloring. For example, consider the vertex marked in yellow. No matter where in the graph you start, if you traverse all nine edges in the walk "blue-red-red—blue-red-red—blue-red-red", you will end up at the yellow vertex.

On the other hand, given a homomorphism G → H between undirected graphs, any orientation of H can be pulled back to an orientation of G so that has a homomorphism to . Therefore, a graph G is k-colorable (has a homomorphism to Kk) if and only if some orientation of G has a homomorphism to k. A folklore theorem states that for all k, a directed graph G has a homomorphism to k if and only if it admits no homomorphism from the directed path k+1. Here n is the directed graph with vertices 1, 2, …, n and edges from i to i + 1, for i = 1, 2, …, n − 1.

In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices with no outgoing edges. That is, the graph should have no edges that start within the closure and end outside the closure. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph... It may be solved in polynomial time using a reduction to the maximum flow problem. It may be used to model various application problems of choosing an optimal subset of tasks to perform, with dependencies between pairs of tasks, one example being in open pit mining.

The first player unable to extend the path loses. An illustration of the game (containing some cities in Michigan) is shown in the figure below. :Image:Generalized geography 1.svg In a generalized geography (GG) game, we replace the graph of city names with an arbitrary directed graph.

Information fuzzy networks (IFN) is a greedy machine learning algorithm for supervised learning. The data structure produced by the learning algorithm is also called Info Fuzzy Network. IFN construction is quite similar to decision trees' construction. However, IFN constructs a directed graph and not a tree.

He is a professor at the Department of Management Science and Technology at the Athens University of Economics and Business, and a member of the IEEE Software editorial board, contributing the Tools of the TradeTools of the Trade column. Since 2014, he is also editor-in-chief of IEEE Software. Spinellis is a four-time winner of the International Obfuscated C Code Contest (1988, 1990, 1991, 1995). He is also a committer in the FreeBSD project, and author of a number of popular free or open-source systems: the UMLGraphUMLGraph declarative UML diagram generator, the bib2xhtmlbib2xhtml BibTeX to XHTML converter, the outwitoutwit Microsoft Windows data with command line programs integration tool suite, the CScoutCScout source code analyzer and refactoring browser, the socketpipesocketpipe fast IPC plumbing utility and directed graph shelldgsh – directed graph shell the directed graph Unix shell for big data and stream processing pipelines.. In 2008, together with a collaborator, Spinellis claimed that "red links" (a Wikipedia slang for wikilinks that lead to non- existing pages) is what drives Wikipedia growth.

Tarjan's algorithm is an algorithm in graph theory for finding the strongly connected components of a directed graph. It runs in linear time, matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm. Tarjan's algorithm is named for its inventor, Robert Tarjan.

Construction of the de Bruijn graphs as iterated line digraphs It is also possible to generalize line graphs to directed graphs. If G is a directed graph, its directed line graph or line digraph has one vertex for each edge of G. Two vertices representing directed edges from u to v and from w to x in G are connected by an edge from uv to wx in the line digraph when v = w. That is, each edge in the line digraph of G represents a length-two directed path in G. The de Bruijn graphs may be formed by repeating this process of forming directed line graphs, starting from a complete directed graph.

An arbitrary directed graph may also be transformed into a DAG, called its condensation, by contracting each of its strongly connected components into a single supervertex.. When the graph is already acyclic, its smallest feedback vertex sets and feedback arc sets are empty, and its condensation is the graph itself.

Middendorf-Ziv (MZ) proposed a growing directed graph modeling biological network dynamics. A prototype is chosen at random and duplicated. The prototype or progenitor node has edges pruned with probability β and edges added with probability α<<β. Based loosely on the undirected protein network model of Sole et al.

An example of CSR representation of a directed graph. Pennant data structure for k=0 to k=3. An example of bag structure with 23 elements. There are some special data structures that parallel BFS can benefit from, such as CSR(Compressed Sparse Row), bag-structure, bitmap and so on.

Minimum s–t cut for directed graph. All edges have equal weights. An s–t cut is a partition of the graph into two parts each containing one of either s or t. Where S \cup T = V , \;\; s \in S , \;\; t \in T , the s–t cut is (S, T) .

A connected component is a maximal connected subgraph of an undirected graph. Each vertex belongs to exactly one connected component, as does each edge. A graph is connected if and only if it has exactly one connected component. The strong components are the maximal strongly connected subgraphs of a directed graph.

The concept is formally defined (Bendeck 2008) as a directed graph, with concepts represented as nodes and semantic similarity relations as edges. The relationships are grouped into relation types. The concepts and relations contain attribute values to evaluate the semantic similarityP. Resnik. Using Information Content to Evaluate Semantic Similarity in a Taxonomy. Proc.

The concept of bramble has also been defined for directed graphs. In a directed graph D, a bramble is a collection of strongly connected subgraphs of D that all touch each other: for every pair of disjoint elements X, Y of the bramble, there must exist an arc in D from X to Y and one from Y to X. The order of a bramble is the smallest size of a hitting set, a set of vertices of D that has a nonempty intersection with each of the elements of the bramble. The bramble number of D is the maximum order of a bramble in D. The bramble number of a directed graph is within a constant factor of its directed treewidth.

Combinator graph reduction is a fundamental implementation technique for functional programming languages, in which a program is converted into a combinator representation which is mapped to a directed graph data structure in computer memory, and program execution then consists of rewriting parts of this graph ("reducing" it) so as to move towards useful results.

T-colorings correspond to homomorphisms into certain infinite graphs. An oriented coloring of a directed graph is a homomorphism into any oriented graph. An L(2,1)-coloring is a homomorphism into the complement of the path graph that is locally injective, meaning it is required to be injective on the neighbourhood of every vertex.

GSS (Graph Style Sheets) is an RDF (Resource Description Framework) vocabulary for representation of data in a model of labeled directed graph. Using it will make a relatively complex data resource modeled in RDF, much easier to understand by declaring simple styling and visibility instructions to be applied on selected resources, literals and properties.

The permanent of a (0,1)-matrix is equal to the number of vertex-disjoint cycle covers of a directed graph with this adjacency matrix. This fact is used in a simplified proof showing that computing the permanent is #P-complete.Ben-Dor, Amir and Halevi, Shai. (1993). "Zero-one permanent is #P-complete, a simpler proof".

Here is what a simple directed graph with three nodes and two links between them looks like which looks like this: Image:DgmlGraph.png The complete XSD schema for DGML is available at . DGML not only allows describing nodes and links in a graph, but also annotating those nodes and links with any user defined property and/or category.

Let G = (V, E) be a directed graph. An Eulerian circuit is a directed closed path which visits each edge exactly once. In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian.

One of the thinking processes in the theory of constraints, a current reality tree (CRT) is a way of analyzing many systems or organizational problems at once. By identifying root causes common to most or all of the problems, a CRT can greatly aid focused improvement of the system. A current reality tree is a directed graph.

Given a directed graph G = (V, E), a path cover is a set of directed paths such that every vertex v ∈ V belongs to at least one path. Note that a path cover may include paths of length 0 (a single vertex)., Section 2.5. A path cover may also refer to a vertex-disjoint path cover, i.e.

In addition, Vobjects may have a number of directed relations to other Vobjects, which allows them to form directed graph data structures. VOS is patent free, and its implementation is Free Software. The primary application focus of VOS is general purpose, multiuser, collaborative 3D virtual environments or virtual reality. The primary designer and author of VOS is Peter Amstutz.

The final step in the data flow reconstruction is the topological sorting of the association graph. The directed graph created in the previous step is topologically sorted to obtain the order in which the actors have modified the data. This inherit order of the actors defines the data flow of the big data pipeline or task.

Both edge and vertex contraction techniques are valuable in proof by induction on the number of vertices or edges in a graph, where it can be assumed that a property holds for all smaller graphs and this can be used to prove the property for the larger graph. Edge contraction is used in the recursive formula for the number of spanning trees of an arbitrary connected graph, and in the recurrence formula for the chromatic polynomial of a simple graph. Contractions are also useful in structures where we wish to simplify a graph by identifying vertices that represent essentially equivalent entities. One of the most common examples is the reduction of a general directed graph to an acyclic directed graph by contracting all of the vertices in each strongly connected component.

The transition graph T_I of an instance I of a problem L is a directed graph. The nodes represent all elements of the finite set of solutions F_L(I) and the edges point from one solution to the neighbor with strictly better cost. Therefore it is an acyclic graph. A sink, which is a node with no outgoing edges, is a local optimum.

A project network is a graph (weighted directed graph) depicting the sequence in which a project's terminal elements are to be completed by showing terminal elements and their dependencies. It is always drawn from left to right to reflect project chronology. Wednesday, 19 December 2018 Image:project network.png The work breakdown structure or the product breakdown structure show the "part-whole" relations.

The aliquot sequence can be represented as a directed graph, G_{n,s}, for a given integer n, where s(k) denotes the sum of the proper divisors of k. Cycles in G_{n,s} represent sociable numbers within the interval [1,n]. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

An aperiodic graph. The cycles in this graph have lengths 5 and 6; therefore, there is no k > 1 that divides all cycle lengths. strongly connected graph with period three. In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph.

USTCON is a special case of STCON (directed reachability), the problem of determining whether a directed path between two vertices in a directed graph exists, which is complete for NL. Because USTCON is SL-complete, most advances that impact USTCON have also impacted SL. Thus they are connected, and discussed together. In October 2004 Omer Reingold showed that SL = L.

They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. Kn can be decomposed into trees Ti such that Ti has vertices.

A signed digraph is a directed graph with signed arcs. Signed digraphs are far more complicated than signed graphs, because only the signs of directed cycles are significant. For instance, there are several definitions of balance, each of which is hard to characterize, in strong contrast with the situation for signed undirected graphs. Signed digraphs should not be confused with oriented signed graphs.

The aliquot sequence can be represented as a directed graph, G_{n,s}, for a given integer n, where s(k) denotes the sum of the proper divisors of k. Cycles in G_{n,s} represent sociable numbers within the interval [1,n]. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

An effective algorithm that can check emptiness of a Büchi automaton: # Consider the automaton as a directed graph and decompose it into strongly connected components (SCCs). # Run a search (e.g., the depth-first search) to find which SCCs are reachable from the initial state. # Check whether there is a non- trivial SCC that is reachable and contains a final state.

The term arborescence comes from French. Some authors object to it on grounds that it is cumbersome to spell. There is a large number of synonyms for arborescence in graph theory, including directed rooted tree out-arborescence, out-tree, and even branching being used to denote the same concept. Rooted tree itself has been defined by some authors as a directed graph.

A vertex in a directed graph whose second neighborhood is at least as large as its first neighborhood is called a Seymour vertex. In the second neighborhood conjecture, the condition that the graph have no two-edge cycles is necessary, for in graphs that have such cycles (for instance the complete oriented graph) all second neighborhoods may be empty or small.

Although her conjecture led to subsequent research in the same area, it has been shown to be inconsistent with the actual average packing density in dimensions two through four. Palásti's results in the theory of random graphs include bounds on the probability that a random graph has a Hamiltonian circuit, and on the probability that a random directed graph is strongly connected.

The theorem relates two quantities: the maximum flow through a network, and the minimum weight of a cut of the network. To state the theorem, each of these quantities must first be defined. Let be a directed graph, where V denotes the set of vertices and E is the set of edges. Let and be the source and the sink of , respectively.

Social network visualization using a force-directed graph drawing algorithm.Visualization of links between pages on a wiki using a force- directed layout. Force-directed graph drawing algorithms are a class of algorithms for drawing graphs in an aesthetically-pleasing way. Their purpose is to position the nodes of a graph in two-dimensional or three-dimensional space so that all the edges are of more or less equal length and there are as few crossing edges as possible, by assigning forces among the set of edges and the set of nodes, based on their relative positions, and then using these forces either to simulate the motion of the edges and nodes or to minimize their energy.. While graph drawing can be a difficult problem, force-directed algorithms, being physical simulations, usually require no special knowledge about graph theory such as planarity.

It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex. Any directed graph may be made into a DAG by removing a feedback vertex set or a feedback arc set, a set of vertices or edges (respectively) that touches all cycles. However, the smallest such set is NP-hard to find., Problems GT7 and GT8, pp. 191–192.

It allows more powerful results, for example on Grushko's theorem, and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph. # Grushko's theorem has the consequence that if a subset B of a free group F on n elements generates F and has n elements, then B generates F freely.

Force-directed graph drawing systems continue to be a popular method for visualizing graphs, but these systems typically use more complicated systems of forces that combine attractive forces on graph edges (as in Tutte's embedding) with repulsive forces between arbitrary pairs of vertices. These additional forces may cause the system to have many locally stable configurations rather than, as in Tutte's embedding, a single global solution..

A discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems, which may also be called analog systems. A final discrete system is often modeled with a directed graph and is analyzed for correctness and complexity according to computational theory. Because discrete systems have a countable number of states, they may be described in precise mathematical models.

Each actor in ROOM has a behavior which is defined by means of a hierarchical finite-state machine, or just state machine for short. A state machine is a directed graph consisting of nodes called states and edges called transitions. State transitions are triggered by incoming messages from an internal or external end port. In this context the messages sometimes are also called events or signals.

Robert E. Tarjan. Other significant results on the design and analysis of data structures were contributed on the problems of interpolation search, negative cycleTsakalidis, Athanasios K.: Finding a Negative Cycle in a Directed Graph. Techn. Report A85/05, Angewandte Mathematik und Informatik, FB-10, Univ. des Saarlandes, Saarbrücken (1985) and nearest common ancestorvan Leeuwen, J., Tsakalidis, A.K.: An optimal Pointer Machine Algorithm for Nearest Common Ancestors. Tech.

In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components.

Please, a Go-based build system, was developed in 2016 by Thought Machine who was also inspired by Google's Bazel and dissatisfied with Facebook's Buck. Bazel, Buck, Pants and Please all use the same Starlark (formerly Skylark) build language, a domain-specific language based on Python. Some specialized monorepo build tools, such as Lerna, solve fetching of duplicate dependencies, but lack any directed graph capabilities.

For each red or blue edge , is reachable from : there exists a blue path starting at and ending at . A vertex of a directed graph is said to be reachable from another vertex when there exists a path that starts at and ends at . As a special case, every vertex is considered to be reachable from itself (by a path with zero edges). If a vertex can reach itself via a nontrivial path (a path with one or more edges), then that path is a cycle, so another way to define directed acyclic graphs is that they are the graphs in which no vertex can reach itself via a nontrivial path.. A topological ordering of a directed graph is an ordering of its vertices into a sequence, such that for every edge the start vertex of the edge occurs earlier in the sequence than the ending vertex of the edge.

Directed graph of relationships among SNP prediction webservers and their bioinformatics sources. Single nucleotide polymorphisms (SNPs) play an important role in genome wide association studies because they act as primary biomarkers. SNPs are currently the marker of choice due to their large numbers in virtually all populations of individuals. The location of these biomarkers can be tremendously important in terms of predicting functional significance, genetic mapping and population genetics.

Cycles of all possible lengths in the graph of an octahedron, showing it to be pancyclic. In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph.. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.

A labeled-property graph model is represented by a set of nodes, relationships, properties, and labels. Both nodes of data and their relationships are named and can store properties represented by key/value pairs. Nodes can be labelled to be grouped. The edges representing the relationships have two qualities: they always have a start node and an end node, and are directed; making the graph a directed graph.

Well-known puzzles fitting this description are mechanical puzzles like Rubik's Cube, Towers of Hanoi, and the 15 puzzle. The one-person game of peg solitaire is also covered, as well as many logic puzzles, such as the missionaries and cannibals problem. These have in common that they can be modeled mathematically as a directed graph, in which the configurations are the vertices, and the moves the arcs.

A quiver is simply a directed graph (with loops and multiple arrows allowed), but it can be made into a category (and also an algebra) by considering paths in the graph. Representations of such categories/algebras have illuminated several aspects of representation theory, for instance by allowing non-semisimple representation theory questions about a group to be reduced in some cases to semisimple representation theory questions about a quiver.

For all primes up to , only in two cases: and , where is the number of vertices in the cycle of 1 in the doubling diagram modulo . Here the doubling diagram represents the directed graph with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2x reduced modulo m. It was shown, that for all odd prime numbers either or .

The Galois group of a polynomial of degree n is S_n or a proper subgroup of that. If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup. Transitive subgroups of S_n form a directed graph: one group can be a subgroup of several groups. One resolvent can tell if the Galois group of a polynomial is a (not necessarily proper) subgroup of given group.

The problem can also be stated in terms of zero-one matrices. The connection can be seen if one realizes that each directed graph has an adjacency matrix where the column sums and row sums correspond to (a_1,\cdots,a_n) and (b_1,\ldots,b_n). Note that the diagonal of the matrix only contains zeros. The problem is then often denoted by 0-1-matrices for given row and column sums.

In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by , as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to a unique set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom.

The order can be described through a history tree which is a directed graph, "because it is possible to continue redoing commands from another branch creating a link in the graph". Even though the set of commands is simple and easy to understand, the complex structure with skipping and linking branches is hard to comprehend and to remember, when the user wants to undo more than one steps.

Let be a weighted directed graph with vertex set and edge set (figure A); let be a designated source vertex in , and let be a designated destination vertex. Let each edge in , from vertex to vertex , have a non-negative cost . Define to be the cost of the shortest path to vertex from vertex in the shortest path tree rooted at (figure C). Note: Node and Vertex are often used interchangeably.

Authors which give the more general definition, may refer to these as connected (or 1-connected) rooted digraphs. The Art of Computer Programming defines rooted digraphs slightly more broadly, namely a directed graph is called rooted if it has at least one node that can reach all the other nodes; Knuth notes that the notion thus defined is a sort of intermediate between the notions of strongly connected and connected digraph.

The Calkin–Wilf tree, drawn using an H tree layout. The Calkin–Wilf tree may be defined as a directed graph in which each positive rational number occurs as a vertex and has one outgoing edge to another vertex, its parent. We assume that is in simplest terms; that is, the greatest common divisor of and is 1. If , the parent of is ; if , the parent of is .

Directed graph showing the orbits of small numbers under the Collatz map. The Collatz conjecture states that all paths eventually lead to 1. The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer . Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term.

A raffinement on the above definition can be made, resulting in the concept of acyclic agreement forest. An agreement forest for two -trees and is said to be acyclic if each of its tree components can be numbered in such a way that if the root of one component is an ancestor of the root of another component in either or , then the number assigned to is lower than the number assigned to . Another characterization of acyclicity in agreement forest is to consider the directed graph that has vertex set and a directed edge if and only if and at least one of the two following conditions hold: # the root of is an ancestor of the root of in # the root of is an ancestor of the root of in The directed graph is called the inheritance graph associated with the agreement forest , and we call acyclic if has no directed cycle.

There are also more subtle limitations of first-order logic that are implied by the compactness theorem. For example, in computer science, many situations can be modeled as a directed graph of states (nodes) and connections (directed edges). Validating such a system may require showing that no "bad" state can be reached from any "good" state. Thus one seeks to determine if the good and bad states are in different connected components of the graph.

A roadmap G = (V,E) is a directed graph. Each vertex v is a randomly sampled conformation in C. Each (directed) edge from vertex vi to vertex vj carries a weight Pij , which represents the probability that the molecule will move to conformation vj , given that it is currently at vi. The probability Pij is 0 if there is no edge from vi to vj. Otherwise, it depends on the energy difference between conformations.

In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths.. Gammoids were given their name by . and studied in more detail by ..

Sambasiva Rao Kosaraju is a professor of computer science at Johns Hopkins University, and division director for Computing & Communication Foundations at the National Science Foundation.Staff Announcement – CCF, Farnam Jahanian, NSF, retrieved 2014-01-14. He has done extensive work in the design and analysis of parallel and sequential algorithms. In 1978, he wrote a paper describing a method to efficiently compute strongly connected members of a directed graph, a method later called Kosaraju's algorithm.

Graph with strongly connected components marked In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)).

In words, when , one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or ≲ is used instead of ≤. To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive.

The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)) to be the indegree- outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. Fulkerson D.R. Fulkerson: Zero-one matrices with zero trace.

The road coloring problem is the problem of labeling the edges of a regular directed graph with the symbols of a k-letter input alphabet (where k is the outdegree of each vertex) in order to form a synchronizable DFA. It was conjectured in 1970 by Benjamin Weiss and Roy Adler that any strongly connected and aperiodic regular digraph can be labeled in this way; their conjecture was proven in 2007 by Avraham Trahtman..

The Kleitman–Wang algorithms are two different algorithms in graph theory solving the digraph realization problem, i.e. the question if there exists for a finite list of nonnegative integer pairs a simple directed graph such that its degree sequence is exactly this list. For a positive answer the list of integer pairs is called digraphic. Both algorithms construct a special solution if one exists or prove that one cannot find a positive answer.

Many topological sorting algorithms will detect cycles too, since those are obstacles for topological order to exist. Also, if a directed graph has been divided into strongly connected components, cycles only exist within the components and not between them, since cycles are strongly connected. For directed graphs, distributed message based algorithms can be used. These algorithms rely on the idea that a message sent by a vertex in a cycle will come back to itself.

The feedback vertex set problem can be solved in polynomial time on graphs of maximum degree at most three.; Note that the problem of deleting as few edges as possible to make the graph cycle-free is equivalent to finding a spanning tree, which can be done in polynomial time. In contrast, the problem of deleting edges from a directed graph to make it acyclic, the feedback arc set problem, is NP-complete.

The scripting language is a dataflow language: a programming paradigm that describes a directed graph of the data flowing between operations. It lacks most procedural programming control structures, but containing many features familiar to programmers, including variables, distinct datatypes, conditionals, and complex expressions. The language works primarily with the audio/video clip as a built-in data type. The clip is a complex structure with many attributes such as width, height and duration.

An abstract dependency graph is a directed graph, a graph of vertices connected by one-way edges. Most often, the vertices and edges of the graph represent the inputs and outputs of functions in or components of the system. By inspecting the created abstract dependency graph, the developer can detect syntactic anomalies (or Preece anomalies) in the system. While anomalies are not always defects, they often provide clues to finding defects in a system.

In the symmetric TSP, the distance between two cities is the same in each opposite direction, forming an undirected graph. This symmetry halves the number of possible solutions. In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph. Traffic collisions, one-way streets, and airfares for cities with different departure and arrival fees are examples of how this symmetry could break down.

The file sources would feed compression filters, the output of the compression filters would feed into a multiplexer that would combine the two inputs and produce a single output. (An example of a multiplexer would be an MPEG transport stream creator.) Finally the multiplexer output feeds into a file sink, which would create a file from the output. GStreamer example of a filter graph. A filter graph in multimedia processing is a directed graph.

A parity game. Circular nodes belong to player 0, rectangular nodes belong to player 1. On the left side is the winning region of player 0, on the right side is the winning region of player 1. A parity game is played on a colored directed graph, where each node has been colored by a priority - one of (usually) finitely many natural numbers. Two players, 0 and 1, move a (single, shared) token along the edges of the graph.

Method implementation In Prograph a method is represented by a series of icons, each icon containing an instructions (or group of them). Within each method the flow of data is represented by lines in a directed graph. Data flows in the top of the diagram, passes through various instructions, and eventually flows back out the bottom (if there is any output). Several features of the Prograph system are evident in this picture of a database sorting operation.

Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric (s reaches t iff t reaches s). The connected components of an undirected graph can be identified in linear time. The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph (which, incidentally, need not be symmetric).

This placement can potentially carry semantic information, whereas in node-link graphics the placement is often arbitrarily generated within constraints for aesthetics, such as during force-directed graph drawing, and may result in apparently informative artifacts. Edges are drawn (vertically) in a darker shade than (horizontal) nodes, creating visual distinction. Additional edges increase the width of the graph. Both ends of a link are represented as a square to reinforce the above effect even at small scales.

A graph with three vertices and three edges. A graph (sometimes called undirected graph for distinguishing from a directed graph, or simple graph for distinguishing from a multigraph)See, for instance, Iyanaga and Kawada, 69 J, p. 234 or Biggs, p. 4. is a pair , where is a set whose elements are called vertices (singular: vertex), and is a set of two-sets (sets with two distinct elements) of vertices, whose elements are called edges (sometimes links or lines).

Modern videogame development platforms such as Unity and Unreal Engine increasingly include node-based editors that can create shaders without the need for actual code; the user is instead presented with a directed graph of connected nodes that allow users to direct various textures, maps, and mathematical functions into output values like the diffuse color, the specular color and intensity, roughness/metalness, height, normal, and so on. Automatic compilation then turns the graph into an actual, compiled shader.

Given a directed graph G = (V, E) and two vertices s and t, we are to find the maximum number of paths from s to t. This problem has several variants: 1\. The paths must be edge-disjoint. This problem can be transformed to a maximum flow problem by constructing a network N = (V, E) from G, with s and t being the source and the sink of N respectively, and assigning each edge a capacity of 1.

The obvious transplant is to consider a digraph rooted by identifying a particular node as root.. See in particular p. 307. However, in computer science, these terms commonly refer to a narrower notion, namely a rooted directed graph is a digraph with a distinguished node r, such that there is a directed path from r to any node other than r.. See in particular p. 122.. See in particular p. 524.. See in particular p. 308.

Degree of branching refers to the number of direct subordinates or children an object has (in graph theory, equivalent to the number of other vertices connected to via outgoing arcs, in a directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: linear and branching. In a linear hierarchy, the maximum degree is 1.

More abstractly, the reachability relation in a DAG forms a partial order, and any finite partial order may be represented by a DAG using reachability. Important polynomial time computational problems on DAGs include topological sorting (computing a topological ordering), construction of the transitive closure and transitive reduction (the largest and smallest DAGs with the same reachability relation, respectively) of sets, and the closure problem, in which the goal is to find a minimum-weight subset of vertices with no edges connecting them to the rest of the graph. Transforming a directed graph with cycles into a DAG by deleting as few vertices or edges as possible (the feedback vertex set and feedback edge set problem, respectively) is an NP-hard problem, but any directed graph can be made into a DAG (its condensation) by contracting each strongly connected component into a single supervertex. The problems of finding shortest paths and longest paths can be solved on DAGs in linear time, in contrast to arbitrary graphs for which shortest path algorithms are slower and longest path problems are NP-hard.

It is also possible to check whether a given directed graph is a DAG in linear time, either by attempting to find a topological ordering and then testing for each edge whether the resulting ordering is validFor depth-first search based topological sorting algorithm, this validity check can be interleaved with the topological sorting algorithm itself; see e.g. . or alternatively, for some topological sorting algorithms, by verifying that the algorithm successfully orders all the vertices without meeting an error condition., pp. 50–51.

A query plan is a directed graph where the nodes are operators and the edges describe the processing flow. Each operator in the query plan encapsulates the semantic of a specific operation, such as filtering or aggregation. In DSMSs that process relational data streams, the operators are equal or similar to the operators of the Relational algebra, so that there are operators for selection, projection, join, and set operations. This operator concept allows the very flexible and versatile processing of a DSMS.

A Citation Network (see also citation graph) is a social network that contains document sources which are by the citations from one document to another. Egghe & Rousseau once (1990, p. 228) explained "when a document di cites a document dj, we can show this by an arrow going from the node representing di to the document representing dj. In this way the documents from a collection D form a directed graph, which is called a 'citation graph' or 'citation network' ".

The graphical representation of a hyperedge may be a box (compared to the edge which is a line) and the representations of its tentacles are lines from the box to the connected nodes. In a directed hypergraph, the tentacles carry labels which are determined by the hyperedge's label. A conventional directed graph can be thought of as a hypergraph with hyperedges each of which has two tentacles. These two tentacles are labelled source and target and usually indicated by an arrow.

In mathematics, a minimum bottleneck spanning tree (MBST) in an undirected graph is a spanning tree in which the most expensive edge is as cheap as possible. A bottleneck edge is the highest weighted edge in a spanning tree. A spanning tree is a minimum bottleneck spanning tree if the graph does not contain a spanning tree with a smaller bottleneck edge weight.Everything about Bottleneck Spanning Tree For a directed graph, a similar problem is known as Minimum Bottleneck Spanning Arborescence (MBSA).

In graph theory, the strongly connected components of a directed graph may be found using an algorithm that uses depth-first search in combination with two stacks, one to keep track of the vertices in the current component and the second to keep track of the current search path.. Versions of this algorithm have been proposed by , , , , and ; of these, Dijkstra's version was the first to achieve linear time.History of Path-based DFS for Strong Components, Harold N. Gabow, accessed 2012-04-24.

Another interesting connection concerns orientations of graphs. An orientation of an undirected graph G is any directed graph obtained by choosing one of the two possible orientations for each edge. An example of an orientation of the complete graph Kk is the transitive tournament k with vertices 1,2,…,k and arcs from i to j whenever i < j. A homomorphism between orientations of graphs G and H yields a homomorphism between the undirected graphs G and H, simply by disregarding the orientations.

It is distinct from the order → on equivalence classes of undirected graphs, but contains it as a suborder. This is because every undirected graph can be thought of as a directed graph where every arc (u,v) appears together with its inverse arc (v,u), and this does not change the definition of homomorphism. The order → for directed graphs is again a distributive lattice and a Heyting algebra, with join and meet operations defined as before. However, it is not dense.

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a. In category theory, a quiver can be understood to be the underlying structure of a category, but without composition or a designation of identity morphisms.

In topological graph theory, the notion of a rooted graph may be extended to consider multiple vertices or multiple edges as roots. The former are sometimes called vertex-rooted graphs in order to distinguish them from edge- rooted graphs in this context. Graphs with multiple nodes designated as roots are also of some interest in combinatorics, in the area of random graphs. These graphs are also called multiply rooted graphs.. The terms rooted directed graph or rooted digraph also see variation in definitions.

Few build tools work well in a monorepo, and flows where builds and continuous integration testing of the entire repository are performed upon check-in will cause performance problems. Directed graph builds systems like Buck, Bazel, Pants and Please solve this by compartmentalizing builds and tests to the active area of development. Twitter began development of Pants in 2011, as both Facebook's Buck and Google's Bazel were closed-source at the time. Twitter open-sourced Pants in 2012 under the Apache 2.0 License.

Any undirected graph may be made into a DAG by choosing a total order for its vertices and directing every edge from the earlier endpoint in the order to the later endpoint. The resulting orientation of the edges is called an acyclic orientation. Different total orders may lead to the same acyclic orientation, so an -vertex graph can have fewer than acyclic orientations. The number of acyclic orientations is equal to , where is the chromatic polynomial of the given graph.. condensation of the blue directed graph.

NodeXL generates an interactive canvas for visualizing graphs. The project allows users to pick from several well-known Force-directed graph drawing layout algorithms such as Fruchterman-Reingold and Harel-Koren. NodeXL allows the user to multi-select, drag and drop nodes on the canvas and to manually edit their visual properties (size, color, and opacity). In addition, NodeXL allows users to map the visual properties of nodes and edges to metrics it calculates, and in general to any column in the edges and vertices worksheet.

The counting version of this problem asks for the number of Hamiltonian cycles in a given directed graph. Seta Takahiro provided a reduction from 3SAT to this problem when restricted to planar directed max degree-3 graphs. The reduction provides a bijection between the solutions to an instance of 3SAT and the solutions to an instance of Hamiltonian Cycle in planar directed max degree-3 graphs. Hence the reduction is parsimonious and Hamiltonian Cycle in planar directed max degree-3 graphs is #P-complete.

For a Hamiltonian decomposition to exist in an undirected graph, the graph must be connected and regular of even degree. A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even. The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph.

Given a directed graph G, the minimum path cover problem consists of finding a path cover for G having the fewest paths. A minimum path cover consists of one path if and only if there is a Hamiltonian path in G. The Hamiltonian path problem is NP-complete, and hence the minimum path cover problem is NP-hard. However, if the graph is acyclic, the problem is in complexity class P and can therefore be solved in polynomial time by transforming it in a matching problem.

A context-free grammar G is an SLG if: 1\. for every non-terminal N, there is at most one production rule that has N as its left-hand side, and 2\. the directed graph G=, defined by V being the set of non-terminals and (A,B) ∈ E whenever B appears at the right-hand side of a production rule for A, is acyclic. A mathematical definition of the more general formalism of straight-line context-free tree grammars can be found in Lohrey et al.

This application was the original motivation for Coffman and Graham to develop their algorithm... In the layered graph drawing framework outlined by . the input is a directed graph, and a drawing of a graph is constructed in several stages:. Bastert and Matuszewski also include a description of the Coffman–Graham algorithm; however, they omit the transitive reduction stage of the algorithm. #A feedback arc set is chosen, and the edges of this set reversed, in order to convert the input into a directed acyclic graph.

Most approaches to behavioral detection are based on analysis of system call dependencies. The executed binary code is traced using strace or more precise taint analysis to compute data-flow dependencies among system calls. The result is a directed graph G=(V,E) such that nodes are system calls, and edges represent dependencies. For example, (s,t)\in E if a result returned by system call s (either directly as a result or indirectly through output parameters) is later used as a parameter of system call t.

A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is unilaterally connected or unilateral (also called semiconnected) if it contains a directed path from to or a directed path from to for every pair of vertices .Chapter 11: Digraphs: Principle of duality for digraphs: Definition It is strongly connected, or simply strong, if it contains a directed path from to and a directed path from to for every pair of vertices .

Pic.2: In this diagram, reduces to both or in zero or more rewrite steps (denoted by the asterisk). In order for the rewrite relation to be confluent, both reducts must in turn reduce to some common . A rewriting system can be expressed as a directed graph in which nodes represent expressions and edges represent rewrites. So, for example, if the expression a can be rewritten into b, then we say that b is a reduct of a (alternatively, a reduces to b, or a is an expansion of b).

The transitive reduction of a DAG is the graph with the fewest edges that represents the same reachability relation as . It is a subgraph of , formed by discarding the edges for which also contains a longer path connecting the same two vertices. Like the transitive closure, the transitive reduction is uniquely defined for DAGs. In contrast, for a directed graph that is not acyclic, there can be more than one minimal subgraph with the same reachability relation.. A Hasse diagram representing the partial order of set inclusion (⊆) among the subsets of a three-element set.

Every hydrogen is bonded to two oxygens, strongly to one and weakly to the other. The resulting configuration is geometrically a periodic lattice. The distribution of bonds on this lattice is represented by a directed-graph (arrows) and can be either ordered or disordered. In 1935, Linus Pauling used the ice rules to calculate the residual entropy (zero temperature entropy) of ice Ih. For this (and other) reasons the rules are sometimes mis-attributed and referred to as "Pauling's ice rules" (not to be confused with Pauling's rules for ionic crystals).

In game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves. The complete game tree for a game is the game tree starting at the initial position and containing all possible moves from each position; the complete tree is the same tree as that obtained from the extensive-form game representation. The first two plies of the game tree for tic-tac-toe. The diagram shows the first two levels, or plies, in the game tree for tic-tac-toe.

In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a website can be represented by a directed graph, in which the vertices represent web pages and directed edges represent links from one page to another. A similar approach can be taken to problems in social media, travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases, and many other fields. The development of algorithms to handle graphs is therefore of major interest in computer science.

A tournament is a directed graph with one directed edge between each pair of vertices. Intuitively, a tournament can be used to model a round-robin sports competition, by drawing an edge from the winner to the loser of each game in the competition. A tournament is called strongly connected or strong if and only if it cannot be partitioned into two nonempty subsets L and W of losers and winners, such that every competitor in W beats every competitor in L., Corollary 5b. Every strong tournament is pancyclic, Theorem 7.

Probabilistic graphical models provide a convenient framework for comparing model-based approximations. In this context, value of the process at index x_k \in X can then be represented by a vertex in a directed graph and edges correspond to the terms in the factorization of the joint density of y(X). In general, when no independece relations are assumed, the joint probability distribution can be represented by an arbitrary directed acyclic graph. Using a particular approximation can then be expressed as a certain way of ordering the vertices and adding or removing specific edges.

In computer science, Kosaraju's algorithm (also known as the Kosaraju–Sharir algorithm) is a linear time algorithm to find the strongly connected components of a directed graph. Aho, Hopcroft and Ullman credit it to S. Rao Kosaraju and Micha Sharir. Kosaraju suggested it in 1978 but did not publish it, while Sharir independently discovered it and published it in 1981. It makes use of the fact that the transpose graph (the same graph with the direction of every edge reversed) has exactly the same strongly connected components as the original graph.

Components are represented as nodes in a workflow. In a mathematical sense, components are modeled as nodes in a directed graph: "pipes" (graph edges) connect components and move data along the from node to node where operations are performed on the data. Users have the choice to use predefined components, or to develop their own. To help in industry-specific applications, such as Next Generation Sequencing (see High-throughput sequencing (HTS) methods), BIOVIA has developed components that greatly reduce the amount of time users need to do common industry-specific tasks.

PPP contains PPAD as a subclass (strict containment is an open problem). This is because End-of-the-Line, which defines PPAD, admits a straightforward polynomial-time reduction to PIGEON. In End-of-the- Line, the input is a start vertex s in a directed graph G where each vertex has at most one successor and at most one predecessor, represented by a polynomial-time computable successor function f. Define a circuit C whose input is a vertex x and whose output is its successor if there is one, or x if it does not.

The initial formulation of the retiming problem as described by Leiserson and Saxe is as follows. Given a directed graph G:=(V,E) whose vertices represent logic gates or combinational delay elements in a circuit, assume there is a directed edge e:=(u,v) between two elements that are connected directly or through one or more registers. Let the weight of each edge w(e) be the number of registers present along edge e in the initial circuit. Let d(v) be the propagation delay through vertex v.

The matroid intersection problem becomes NP-hard when three matroids are involved, instead of only two. One proof of this hardness result uses a reduction from the Hamiltonian path problem in directed graphs. Given a directed graph G with n vertices, and specified nodes s and t, the Hamiltonian path problem is the problem of determining whether there exists a simple path of length n − 1 that starts at s and ends at t. It may be assumed without loss of generality that s has no incoming edges and t has no outgoing edges.

The "frame", or view, can be shared amongst different users of the software. Even with the same underlying information, it's possible for two users to interact around different blueprints or different frames. All data is stored in "u-forms", which are a universal property list (name-value pairs) that can contain basic types, arrays, and links to other u-forms. This directed graph of u-forms forms the basis for all data in the system, including visualized data, blueprints, as well as the frame locations and clipping states.

Every Coxeter diagram has a corresponding Schläfli matrix (so named after Ludwig Schläfli), with matrix elements where p is the branch order between the pairs of mirrors. As a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the limited cases of p = 2,3,4, and 6, which are NOT symmetric in general.

In artificial intelligence and related fields, an argumentation framework, argumentation system or argumentation graph, is a way to deal with contentious information and draw conclusions from it. In an abstract argumentation framework,See Dung (1995) entry-level information is a set of abstract arguments that, for instance, represent data or a proposition. Conflicts between arguments are represented by a binary relation on the set of arguments. In concrete terms, you represent an argumentation framework with a directed graph such that the nodes are the arguments, and the arrows represent the attack relation.

Discrete event simulation Continuous simulation Continuous simulation must be clearly differentiated from discrete and discrete event simulation. Discrete simulation relies upon countable phenomena like the number of individuals in a group, the number of darts thrown, or the number of nodes in a Directed graph. Discrete event simulation produces a system which changes its behaviour only in response to specific events and typically models changes to a system resulting from a finite number of events distributed over time. A continuous simulation applies a Continuous function using Real numbers to represent a continuously changing system.

In the field of computer science, a pre-topological order or pre-topological ordering of a directed graph is a linear ordering of its vertices such that if there is a directed path from vertex u to vertex v and v comes before u in the ordering, then there is also a directed path from vertex v to vertex u. If the graph is a directed acyclic graph (DAG), topological orderings are pre- topological orderings and vice versa. In other cases, any pre-topological ordering gives a partial order.

In computer science and graph theory, the zero-weight cycle problem is the problem of deciding whether a directed graph with weights on the edges (which may be positive or negative or zero) has a cycle in which the sum of weights is 0\. A related problem is to decide whether the graph has a cycle in which the sum of weights is less than 0. This related problem can be solved in polynomial time using the Bellman–Ford algorithm. In contrast, detecting a cycle of weight exactly 0 is an NP-complete problem.

In software testing, a cause–effect graph is a directed graph that maps a set of causes to a set of effects. The causes may be thought of as the input to the program, and the effects may be thought of as the output. Usually the graph shows the nodes representing the causes on the left side and the nodes representing the effects on the right side. There may be intermediate nodes in between that combine inputs using logical operators such as AND and OR. Constraints may be added to the causes and effects.

The road coloring problem is the problem of edge-coloring a directed graph with uniform out- degrees, in such a way that the resulting automaton has a synchronizing word. solved the road coloring problem by proving that such a coloring can be found whenever the given graph is strongly connected and aperiodic. Ramsey's theorem concerns the problem of -coloring the edges of a large complete graph in order to avoid creating monochromatic complete subgraphs of some given size . According to the theorem, there exists a number such that, whenever , such a coloring is not possible.

Two edges of a graph are called adjacent if they share a common vertex. Two edges of a directed graph are called consecutive if the head of the first one is the tail of the second one. Similarly, two vertices are called adjacent if they share a common edge (consecutive if the first one is the tail and the second one is the head of an edge), in which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident.

Second, the graph theory term digraph (a portmanteau from directed graph) is defined as, "A graph in which each line has a direction associated with it; a finite, non-empty set of elements together with a set of ordered pairs of these elements." The two digraph terms were first recorded in 1788 and 1955, respectively. The OED2 defines two digraphic meanings, "Pertaining to or of the nature of a digraph" and "Written in two different characters or alphabets." It gives their earliest examples in 1873 and 1880 (which was used meaning "digraphia").

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.

A directed cycle graph of length 8 A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. A directed cycle graph has uniform in-degree 1 and uniform out-degree 1\.

Given a DFA, the problem of determining if it has a synchronizing word can be solved in polynomial time using a theorem due to Ján Černý. A simple approach considers the power set of states of the DFA, and builds a directed graph where nodes belong to the power set, and a directed edge describes the action of the transition function. A path from the node of all states to a singleton state shows the existence of a synchronizing word. This algorithm is exponential in the number of states.

The original coolstreaming code is developed with Python 2.3 on Windows. Coolstreaming is a data-centric design of peer-to-peer streaming overlay. Notable features of the protocol include its intelligent scheduling algorithm that copes well with the bandwidth differences of uploading clients and thus minimises skipping during playback, and its swarm-style architecture that uses a directed graph based on gossip algorithms to broadcast content availability. Coolstreaming is the first P2PTV system that attracted a remarkable number of clients (over one million, while most of the previous systems attracted less than one thousand clients).

Schedule compliance with conflict serializability can be tested with the precedence graph (serializability graph, serialization graph, conflict graph) for committed transactions of the schedule. It is the directed graph representing precedence of transactions in the schedule, as reflected by precedence of conflicting operations in the transactions. :In the precedence graph transactions are nodes and precedence relations are directed edges. There exists an edge from a first transaction to a second transaction, if the second transaction is in conflict with the first (see Conflict serializability above), and the conflict is materialized (i.e.

The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph. :Rahman-Kaykobad (2005). A simple graph with n vertices has a Hamiltonian path if, for every non-adjacent vertex pairs the sum of their degrees and their shortest path length is greater than n. The above theorem can only recognize the existence of a Hamiltonian path in a graph and not a Hamiltonian Cycle.

A special case of interest are rooted trees, the trees with a distinguished root vertex. If the directed paths from the root in the rooted digraph are additionally restricted to be unique, then the notion obtained is that of (rooted) arborescence—the directed-graph equivalent of a rooted tree. A rooted graph contains an arborescence with the same root if and only if the whole graph can be reached from the root, and computer scientists have studied algorithmic problems of finding optimal arborescences.. Rooted graphs may be combined using the rooted product of graphs.

Equivalently, a DAG is a directed graph that has a topological ordering, a sequence of the vertices such that every edge is directed from earlier to later in the sequence. DAGs can model many different kinds of information. For example, a spreadsheet can be modeled as a DAG, with a vertex for each cell and an edge whenever the formula in one cell uses the value from another; a topological ordering of this DAG can be used to update all cell values when the spreadsheet is changed. Similarly, topological orderings of DAGs can be used to order the compilation operations in a makefile.

The Shortest Path Faster Algorithm (SPFA) is an improvement of the Bellman–Ford algorithm which computes single-source shortest paths in a weighted directed graph. The algorithm is believed to work well on random sparse graphs and is particularly suitable for graphs that contain negative- weight edges.About the so-called SPFA algorithm However, the worst-case complexity of SPFA is the same as that of Bellman–Ford, so for graphs with nonnegative edge weights Dijkstra's algorithm is preferred. The SPFA algorithm was first published by Edward F. Moore in 1959, as a generalization of breadth first search; SPFA is Moore's “Algorithm D”.

Retiming is the technique of moving the structural location of latches or registers in a digital circuit to improve its performance, area, and/or power characteristics in such a way that preserves its functional behavior at its outputs. Retiming was first described by Charles E. Leiserson and James B. Saxe in 1983. The technique uses a directed graph where the vertices represent asynchronous combinational blocks and the directed edges represent a series of registers or latches (the number of registers or latches can be zero). Each vertex has a value corresponding to the delay through the combinational circuit it represents.

For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix A of the graph, if Q is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries are unity except the diagonal entries which are zero), then the adjacency matrix of the complement of A is Q-A. The complement is not defined for multigraphs.

In information science and bibliometrics, a citation graph (or citation network) is a directed graph in which each vertex represents a document and in which each edge represents a citation from the current publication to another. The best known example is probably the citation graph where academic papers are the vertices, as described in the classic 1965 article "Networks of Scientific Papers" by Derek J. de Solla Price. Another example is formed by court judgements in which judges refer to earlier judgements to support their decisions. Citation analysis in a legal context is therefore an important commercial field.

Proceedings of 17-th IFAC World Congress, Seoul, Korea 05 – 12 July 2008 The computer program generates a number of mathematical equations that satisfy given restrictions. Then the optimization algorithm finds the structure of appropriate mathematical expression and its parameters. Fig. 1. Graph for Expression 1 Network operator is a directed graph that corresponds to some mathematical expressions. Every source nodes of the graph are variables or constants of mathematical expression, inner nodes correspond to binary operations and edges correspond to unary operations. The calculation’s result of mathematical expression is kept in the last sink node.

Technically, there is a fourth type, Read after Read (RAR or "Input"): Both instructions read the same location. Input dependence does not constrain the execution order of two statements, but it is useful in scalar replacement of array elements. To make sure we respect the three types of dependencies, we construct a dependency graph, which is a directed graph where each vertex is an instruction and there is an edge from I1 to I2 if I1 must come before I2 due to a dependency. If loop-carried dependencies are left out, the dependency graph is a directed acyclic graph.

Terms involving connected are also used for properties that are related to, but clearly different from, connectedness. For example, a path-connected topological space is simply connected if each loop (path from a point to itself) in it is contractible; that is, intuitively, if there is essentially only one way to get from any point to any other point. Thus, a sphere and a disk are each simply connected, while a torus is not. As another example, a directed graph is strongly connected if each ordered pair of vertices is joined by a directed path (that is, one that "follows the arrows").

A recurrent neural network (RNN) is a class of artificial neural networks where connections between nodes form a directed graph along a temporal sequence. This allows it to exhibit temporal dynamic behavior. Derived from feedforward neural networks, RNNs can use their internal state (memory) to process variable length sequences of inputs. This makes them applicable to tasks such as unsegmented, connected handwriting recognition or speech recognition. The term “recurrent neural network” is used indiscriminately to refer to two broad classes of networks with a similar general structure, where one is finite impulse and the other is infinite impulse.

For a directed graph, Camerini's algorithm focuses on finding the set of edges that would have its maximum cost as the bottleneck cost of the MBSA. This is done by partitioning the set of edges E into two sets A and B and maintaining the set T that is the set in which it is known that GT does not have a spanning arborescence, increasing T by B whenever the maximal arborescence of G(B ∪ T) is not a spanning arborescence of G, otherwise we decrease E by A. The total time complexity is O(E log E).

An oriented coloring can exist only for a directed graph with no loops or directed 2-cycles. For, a loop cannot have different colors at its endpoints, and a 2-cycle cannot have both of its edges consistently oriented between the same two colors. If these conditions are satisfied, then there always exists an oriented coloring, for instance the coloring that assigns a different color to each vertex. If an oriented coloring is complete, in the sense that no two colors can be merged to produce a coloring with fewer colors, then it corresponds uniquely to a graph homomorphism into a tournament.

In computer programming, dataflow programming is a programming paradigm that models a program as a directed graph of the data flowing between operations, thus implementing dataflow principles and architecture. Dataflow programming languages share some features of functional languages, and were generally developed in order to bring some functional concepts to a language more suitable for numeric processing. Some authors use the term datastream instead of dataflow to avoid confusion with dataflow computing or dataflow architecture, based on an indeterministic machine paradigm. Dataflow programming was pioneered by Jack Dennis and his graduate students at MIT in the 1960s.

A graph with edges colored to illustrate path H-A-B (green), closed path or walk with a repeated vertex B-D-E-F-D-C-B (blue) and a cycle with no repeated edge or vertex H-D-G-H (red). In graph theory, a cycle in a graph is a non- empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices. A graph without cycles is called an acyclic graph.

In floorplanning, the model of a floorplan of an integrated circuit is a set of isothetic rectangles called "blocks" within a larger rectangle called "boundary" (e.g., "chip boundary", "cell boundary"). A possible definition of constraint graphs is as follows. The constraint graph for a given floorplan is a directed graph with vertex set being the set of floorplan blocks and there is an edge from block b1 to b2 (called horizontal constraint), if b1 is completely to the left of b2 and there is an edge from block b1 to b2 (called vertical constraint), if b1 is completely below b2.

A Hamiltonian cycle around a network of six vertices In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron.

Let G=(V,E,w) be a directed Graph with the set of nodes V and the set of edges E\subseteq V\times V. Each edge e \in E has a weight w(e) assigned. The goal of the all-pair-shortest-paths problem is to find the shortest path between all pairs of nodes of the graph. For this path to be unique it is required that the graph does not contain cycles with a negative weight. In the remainder of the article it is assumed that the graph is represented using an adjacency matrix.

Evolutionary graph theory is an area of research lying at the intersection of graph theory, probability theory, and mathematical biology. Evolutionary graph theory is an approach to studying how topology affects evolution of a population. That the underlying topology can substantially affect the results of the evolutionary process is seen most clearly in a paper by Erez Lieberman, Christoph Hauert and Martin Nowak. In evolutionary graph theory, individuals occupy vertices of a weighted directed graph and the weight wi j of an edge from vertex i to vertex j denotes the probability of i replacing j.

These models relate endogenous nodal variables with or without exogenous inputs, under sparsity and low-rank constraints. Multilayer graphs, as well as evolving graphs with memory (such as those emerging with generally nonlinear structural vector autoregressive models) are viewed as exogenous inputs. If the latter are not available, results of Giannakis' team show how to "blindly" identify directed graph topologies by decomposing tensor statistics of nodal data obtained under dynamic graph changes. They further employed such graphs as prior information to offer a unifying graph kernel- based approach to statistical inference of (non) stationary processes over graphs.

Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment.

In general, fractional cascading begins with a catalog graph, a directed graph in which each vertex is labeled with an ordered list. A query in this data structure consists of a path in the graph and a query value q; the data structure must determine the position of q in each of the ordered lists associated with the vertices of the path. For the simple example above, the catalog graph is itself a path, with just four nodes. It is possible for later vertices in the path to be determined dynamically as part of a query, in response to the results found by the searches in earlier parts of the path.

A nearest neighbor graph of 100 points in the Euclidean plane. The nearest neighbor graph (NNG) for a set of n objects P in a metric space (e.g., for a set of points in the plane with Euclidean distance) is a directed graph with P being its vertex set and with a directed edge from p to q whenever q is a nearest neighbor of p (i.e., the distance from p to q is no larger than from p to any other object from P). In many discussions the directions of the edges are ignored and the NNG is defined as an ordinary (undirected) graph.

More formally, a spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group and whose vertices are associated with intertwiners of the edge representations adjacent to it. A spin network, immersed into a manifold, can be used to define a functional on the space of connections on this manifold. One computes holonomies of the connection along every link (closed path) of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way. A remarkable feature of the resulting functional is that it is invariant under local gauge transformations.

Q-systems are a method of directed graph transformations according to given grammar rules, developed at the Université de Montréal by Alain Colmerauer in 1967--70 for use in natural language processing. The Université de Montréal's machine translation system, TAUM-73, used the Q-Systems as its language formalism. The data structure manipulated by a Q-system is a Q-graph, which is a directed acyclic graph with one entry node and one exit node, where each arc bears a labelled ordered tree. An input sentence is usually represented by a linear Q-graph where each arc bears a word (tree reduced to one node labelled by this word).

The Rado graph was first constructed by in two ways, with vertices either the hereditarily finite sets or the natural numbers. (Strictly speaking Ackermann described a directed graph, and the Rado graph is the corresponding undirected graph given by forgetting the directions on the edges.) constructed the Rado graph as the random graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. rediscovered the Rado graph as a universal graph, and gave an explicit construction of it with vertex set the natural numbers.

Let G be a directed graph, S be a set of starting vertices, and T be a set of destination vertices (not necessarily disjoint from S). The gammoid \Gamma derived from this data has T as its set of elements. A subset I of T is independent in \Gamma if there exists a set of vertex-disjoint paths whose starting points all belong to S and whose ending points are exactly I.. A strict gammoid is a gammoid in which the set T of destination vertices consists of every vertex in G. Thus, a gammoid is a restriction of a strict gammoid, to a subset of its elements.

Let G=(V,A) be a directed graph and v be a vertex. The set of nodes accessible from v can be defined as the set S which is the least fixed-point for the property: v belongs to S and if w belongs to S and there is an edge from w to x, then x belongs to S. The set of nodes which are co- accessible from v is defined by a similar least fix-point. On the one hand the strongly connected component of v is the intersection of those two least fixed-point. Let G be a proper context-free grammar.

The algorithm takes a directed graph as input, and produces a partition of the graph's vertices into the graph's strongly connected components. Each vertex of the graph appears in exactly one of the strongly connected components. Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, a vertex whose in-degree or out-degree is 0, or any vertex of an acyclic graph. The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on any nodes that have not yet been found).

A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of directed graphs (also called digraphs), which includes as well that of oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications. i SFGs are most commonly used to represent signal flow in a physical system and its controller(s), forming a cyber- physical system.

Phylogenetic trees generated by computational phylogenetics can be either rooted or unrooted depending on the input data and the algorithm used. A rooted tree is a directed graph that explicitly identifies a most recent common ancestor (MRCA), usually an imputed sequence that is not represented in the input. Genetic distance measures can be used to plot a tree with the input sequences as leaf nodes and their distances from the root proportional to their genetic distance from the hypothesized MRCA. Identification of a root usually requires the inclusion in the input data of at least one "outgroup" known to be only distantly related to the sequences of interest.

A bioinformatics workflow management system is a specialized form of workflow management system designed specifically to compose and execute a series of computational or data manipulation steps, or a workflow, that relate to bioinformatics. There are currently many different workflow systems. Some have been developed more generally as scientific workflow systems for use by scientists from many different disciplines like astronomy and earth science. All such systems are based on an abstract representation of how a computation proceeds in the form of a directed graph, where each node represents a task to be executed and edges represent either data flow or execution dependencies between different tasks.

In mathematics, the Coates graph or Coates flow graph, named after C.L. Coates, is a graph associated with the Coates' method for the solution of a system of linear equations. The Coates graph Gc(A) associated with an n × n matrix A is an n-node, weighted, labeled, directed graph. The nodes, labeled 1 through n, are each associated with the corresponding row/column of A. If entry aji ≠ 0 then there is a directed edge from node i to node j with weight aji. In other words, the Coates graph for matrix A is the one whose adjacency matrix is the transpose of A.

The definition and properties of Eulerian trails, cycles and graphs are valid for multigraphs as well. An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v, the indegree of v equals the outdegree of v. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour.. Every Eulerian orientation of a connected graph is a strong orientation, an orientation that makes the resulting directed graph strongly connected.

Barbarossa, together with his students, derived the uncertainty principle for signals defined over a graph and established the fundamental correspondence between uncertainty principle and sampling theory for graph signals. He proposed a new definition of the Fourier Transform for signals defined over a directed graph. He derived an analytic model for the eigenfunctions of linear time-varying systems and invented the product high- order ambiguity function, an algorithm useful to estimate the parameters of multi-component polynomial-phase signals. Barbarossa invented new ways to estimate the instantaneous frequency of continuous-phase signals embedded in noise, based on pattern analysis of their time-frequency representation.

The following example illustrates how a Boolean network can model a GRN together with its gene products (the outputs) and the substances from the environment that affect it (the inputs). Stuart Kauffman was amongst the first biologists to use the metaphor of Boolean networks to model genetic regulatory networks. # Each gene, each input, and each output is represented by a node in a directed graph in which there is an arrow from one node to another if and only if there is a causal link between the two nodes. # Each node in the graph can be in one of two states: on or off.

In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactly one outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to the root vertex. If T is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in T, and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formed by removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the empty sequence.

Let be any finite set, be any function from to itself, and be any element of . For any , let . Let be the smallest index such that the value reappears infinitely often within the sequence of values , and let (the loop length) be the smallest positive integer such that . The cycle detection problem is the task of finding and .. One can view the same problem graph-theoretically, by constructing a functional graph (that is, a directed graph in which each vertex has a single outgoing edge) the vertices of which are the elements of and the edges of which map an element to the corresponding function value, as shown in the figure.

One important NL-complete problem is ST-connectivity (or "Reachability") (Papadimitriou 1994 Thrm. 16.2), the problem of determining whether, given a directed graph G and two nodes s and t on that graph, there is a path from s to t. ST-connectivity can be seen to be in NL, because we start at the node s and nondeterministically walk to every other reachable node. ST-connectivity can be seen to be NL-hard by considering the computation state graph of any other NL algorithm, and considering that the other algorithm will accept if and only if there is a (nondetermistic) path from the starting state to an accepting state.

The map is "a navigational structure which supports the dynamic selection of the intention to be achieved next and the appropriate strategy to achieve it"; it is "a process model in which a nondeterministic ordering of intentions and strategies has been included. It is a labelled directed graph with intentions as nodes and strategies as edges between intentions. The directed nature of the graph shows which intentions can follow which one." The map of the CREWS-L'Ecritoire method looks as follow: Process model of the CREWS-L'Ecritoire method The map consists of goals / intentions (marked with ovals) which are connected by strategies (symbolized through arrows).

In graph theory, an interval I(h) in a directed graph is a maximal, single entry subgraph in which h is the only entry to I(h) and all closed paths in I(h) contain h. Intervals were described in 1970 by F. E. Allen and J. Cocke. Interval graphs are integral to some algorithms used in compilers, specifically data flow analyses. The following algorithm finds all the intervals in a graph consisting of vertices N and the entry vertex n0, and with the functions `pred(n)` and `succ(n)` which return the list of predecessors and successors of a given node n, respectively.

The all-pairs widest path problem has applications in the Schulze method for choosing a winner in multiway elections in which voters rank the candidates in preference order. The Schulze method constructs a complete directed graph in which the vertices represent the candidates and every two vertices are connected by an edge. Each edge is directed from the winner to the loser of a pairwise contest between the two candidates it connects, and is labeled with the margin of victory of that contest. Then the method computes widest paths between all pairs of vertices, and the winner is the candidate whose vertex has wider paths to each opponent than vice versa.

The Rocha–Thatte algorithm is a general algorithm for detecting cycles in a directed graph G by message passing among its vertices, based on the bulk synchronous message passing abstraction. This is a vertex-centric approach in which the vertices of the graph work together for detecting cycles. The bulk synchronous parallel model consists of a sequence of iterations, in each of which a vertex can receive messages sent by other vertices in the previous iteration, and send messages to other vertices. In each pass, each active vertex of G sends a set of sequences of vertices to its out-neighbours as described next.

Ten bits have more () states than three decimal digits (). bits are more than sufficient to represent an information (a number or anything else) that requires decimal digits, so information contained in discrete variables with 3, 4, 5, 6, 7, 8, 9, 10… states can be ever superseded by allocating two, three, or four times more bits. So, the use of any other small number than 2 does not provide an advantage. A Hasse diagram: representation of a Boolean algebra as a directed graph Moreover, Boolean algebra provides a convenient mathematical structure for collection of bits, with a semantic of a collection of propositional variables.

The class is formally defined by specifying one of its complete problems, known as End-Of-The-Line: :G is a (possibly exponentially large) directed graph with no isolated vertices, and with every vertex having at most one predecessor and one successor. G is specified by giving a polynomial-time computable function f(v) (polynomial in the size of v) that returns the predecessor and successor (if they exist) of the vertex v. Given a vertex s in G with no predecessor, find a vertex t≠s with no predecessor or no successor. (The input to the problem is the source vertex s and the function f(v)).

A multi-input, multi-output system represented as a noncommutative matrix signal-flow graph. In automata theory and control theory, branches of mathematics, theoretical computer science and systems engineering, a noncommutative signal-flow graph is a tool for modeling interconnected systems and state machines by mapping the edges of a directed graph to a ring or semiring. A single edge weight might represent an array of impulse responses of a complex system (see figure to the right), or a character from an alphabet picked off the input tape of a finite automaton, while the graph might represent the flow of information or state transitions. As diverse as these applications are, they share much of the same underlying theory.

One iteration of the middle-square method, showing a six digit seed, which is then squared, and the resulting value has its middle six digits as the output value (and also as the next seed for the sequence). Directed graph of all 100 2-digit pseudorandom numbers obtained using the middle-square method with n = 2. In mathematics, the middle-square method is a method of generating pseudorandom numbers. In practice it is not a good method, since its period is usually very short and it has some severe weaknesses; repeated enough times, the middle-square method will either begin repeatedly generating the same number or cycle to a previous number in the sequence and loop indefinitely.

More classic lines of evidence cited among supporters of Bayesian inference include conservatism, or the phenomenon where people modify previous beliefs toward, but not all the way to, a conclusion implied by previous observations. This pattern of behavior is similar to the pattern of posterior probability distributions when a Bayesian model is conditioned on data, though critics argued that this evidence had been overstated and lacked mathematical rigor. Alison Gopnik more recently tackled the problem by advocating the use of Bayesian networks, or directed graph representations of conditional dependencies. In a Bayesian network, edge weights are conditional dependency strengths that are updated in light of new data, and nodes are observed variables.

Example: Block diagram and two equivalent signal-flow graph representations. For some authors, a linear signal-flow graph is more constrained than a block diagram, "A signal flow graph may be regarded as a simplified version of a block diagram. ... for cause and effect ... of linear systems ...we may regard the signal-flow graphs to be constrained by more rigid mathematical rules, whereas the usage of the block-diagram notation is less stringent." in that the SFG rigorously describes linear algebraic equations represented by a directed graph. For other authors, linear block diagrams and linear signal-flow graphs are equivalent ways of depicting a system, and either can be used to solve the gain.

NLTSS development started about the same time LTSS was ported to the Cray-1 to become the Cray Time Sharing System. To stay backward compatible with the many scientific applications at LLNL, NLTSS was forced to emulate the prior LTSS operating system's system calls. This emulation was implemented in the form of a compatibility library called "baselib". As one example, while the directory structure and thus the process structure for NLTSS was naturally a directed graph (process capabilities could be stored in directories just like file capabilities or directory capabilities), the baselib library emulated a simple linear (controller – controllee) process structure (not even a tree structure as in Unix) to stay compatible with the previous LTSS.

In general, a distance matrix is a weighted adjacency matrix of some graph. In a network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes.Frank Harary, Robert Z. Norman and Dorwin Cartwright (1965) Structural Models: An Introduction to the Theory of Directed Graphs, pages 134–8, John Wiley & Sons This distance function, while well defined, is not a metric. There need be no restrictions on the weights other than the need to be able to combine and compare them, so negative weights are used in some applications.

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century, in the work of Ramon Llull.. Such a drawing is sometimes referred to as a mystic rose..

The set of homomorphisms from to can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph are in bijection with the graph homomorphisms from to the multigraph definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of as the multigraph , called the power object of . What is special about a multigraph as an algebra is that its operations are unary.

A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with previous subgraphs. According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected. One way to prove this result is to find an ear decomposition of the underlying undirected graph and then orient each ear consistently..

Let G be a finite, strongly connected, directed graph where all the vertices have the same out- degree k. Let A be the alphabet containing the letters 1, ..., k. A synchronizing coloring (also known as a collapsible coloring) in G is a labeling of the edges in G with letters from A such that (1) each vertex has exactly one outgoing edge with a given label and (2) for every vertex v in the graph, there exists a word w over A such that all paths in G corresponding to w terminate at v. The terminology synchronizing coloring is due to the relation between this notion and that of a synchronizing word in finite automata theory.

The state diagram for a Mealy machine associates an output value with each transition edge, in contrast to the state diagram for a Moore machine, which associates an output value with each state. When the input and output alphabet are both , one can also associate to a Mealy Automata an Helix directed graph .Akhavi et al (2012) This graph has as vertices the couples of state and letters, every nodes are of out-degree one, and the successor of is the next state of the automata and the letter that the automata output when it is instate and it reads letter . This graph is a union of disjoint cycles if the automaton is bireversible.

To compute the widest path widths for all pairs of nodes in a dense directed graph, such as the ones that arise in the voting application, the asymptotically fastest known approach takes time where ω is the exponent for fast matrix multiplication. Using the best known algorithms for matrix multiplication, this time bound becomes .. For an earlier algorithm that also used fast matrix multiplication to speed up all pairs widest paths, see and Chapter 5 of Instead, the reference implementation for the Schulze method uses a modified version of the simpler Floyd–Warshall algorithm, which takes time. For sparse graphs, it may be more efficient to repeatedly apply a single-source widest path algorithm.

In some applications, such cycles are undesirable, and we wish to eliminate them and obtain a directed acyclic graph (DAG). One way to do this is simply to drop edges from the graph to break the cycles. Closely related are the feedback vertex set, which is a set of vertices containing at least one vertex from every cycle in the directed graph, and the minimum spanning tree, which is the undirected variant of the feedback arc set problem. A minimal feedback arc set (one that can not be reduced in size by removing any edges) has the additional property that, if the edges in it are reversed rather than removed, then the graph remains acyclic.

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\.

Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted, directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using the Bellman–Ford algorithm to compute a transformation of the input graph that removes all negative weights, allowing Dijkstra's algorithm to be used on the transformed graph.. Section 25.3, "Johnson's algorithm for sparse graphs", pp. 636–640.. It is named after Donald B. Johnson, who first published the technique in 1977.. A similar reweighting technique is also used in Suurballe's algorithm for finding two disjoint paths of minimum total length between the same two vertices in a graph with non-negative edge weights..

Alternatively, they may be expressed as a special type of directed graph, the implication graph, which expresses the variables of an instance and their negations as vertices in a graph, and constraints on pairs of variables as directed edges. Both of these kinds of inputs may be solved in linear time, either by a method based on backtracking or by using the strongly connected components of the implication graph. Resolution, a method for combining pairs of constraints to make additional valid constraints, also leads to a polynomial time solution. The 2-satisfiability problems provide one of two major subclasses of the conjunctive normal form formulas that can be solved in polynomial time; the other of the two subclasses is Horn-satisfiability.

In constructing matchings in undirected graphs, it is important to find alternating paths, paths of vertices that start and end at unmatched vertices, in which the edges at odd positions in the path are not part of a given partial matching and in which the edges at even positions in the path are part of the matching. By removing the matched edges of such a path from a matching, and adding the unmatched edges, one can increase the size of the matching. Similarly, cycles that alternate between matched and unmatched edges are of importance in weighted matching problems. As showed, an alternating path or cycle in an undirected graph may be modeled as a regular path or cycle in a skew-symmetric directed graph.

Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1X, 1Y and 1Z, if explicitly represented, would appear as three arrows, from the letters X, Y, and Z to themselves, respectively.) Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups.

The World Editor tool includes a graphical script editor designed to be accessible to artists and level designers as well as programmers. The script editor allows the user to place various "methods" into a directed graph connected by "fibers" representing action dependencies and the order of execution. Scripts support loops through the creation of cycles in the graph structure, and conditional execution is supported by marking fibers to be followed or not followed based on the result value output by the methods at which they start. The engine ships with several standard script methods that perform simple actions such as enabling or disabling a scene node (for example, to turn a light on or off) and more complex actions such as evaluating an arbitrary mathematical expression.

The existence of well-balanced orientations, together with Menger's theorem, immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least two edge- disjoint paths between every pair of vertices, from which it follows that any well-balanced orientation must be strongly connected. More generally this result implies that every -edge-connected undirected graph can be oriented to form a -edge-connected directed graph. A totally cyclic orientation of a graph is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of becomes strongly connected.

A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipathGraph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, p.205) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.

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.

PPAD is a subset of the class TFNP, the class of function problems in FNP that are guaranteed to be total. The TFNP formal definition is given as follows: :A binary relation P(x,y) is in TFNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y, and for every x, there exists a y such that P(x,y) holds. Subclasses of TFNP are defined based on the type of mathematical proof used to prove that a solution always exists. Informally, PPAD is the subclass of TFNP where the guarantee that there exists a y such that P(x,y) holds is based on a parity argument on a directed graph.

Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in J). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph G. If we let J be the free category generated by G, there is a universal diagram F : J → C whose image contains G. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms. Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.

Semi-transitive orientations provide a powerful tool to study word-representable graphs. A directed graph is semi-transitively oriented iff it is acyclic and for any directed path u1→u2→ ...→ut, t ≥ 2, either there is no edge from u1 to ut or all edges ui → uj exist for 1 ≤ i < j ≤ t. A key theorem in the theory of word-representable graphs states that a graph is word-representable iff it admits a semi-transitive orientation . As a corollary to the proof of the key theorem one obtain an upper bound on word-representants: Each non-complete word-representable graph G is 2(n − κ(G))-representable, where κ(G) is the size of a maximal clique in G .

Both checking whether a 2-coloring is valid and checking whether a given odd-length sequence of vertices is a cycle may be performed more simply than testing bipartiteness. Analogously, it is possible to test whether a given directed graph is acyclic by a certifying algorithm that outputs either a topological order or a directed cycle. It is possible to test whether an undirected graph is a chordal graph by a certifying algorithm that outputs either an elimination ordering (an ordering of all vertices such that, for every vertex, the neighbors that are later in the ordering form a clique) or a chordless cycle. And it is possible to test whether a graph is planar by a certifying algorithm that outputs either a planar embedding or a Kuratowski subgraph.

October 1972 and graphs (AMBIT-G)Carlos Christensen: An Example of the Manipulation of Directed Graphs in the AMBIT/G Programming Language, in Melvin Klerer et al: Interactive Systems for Experimental Applied Mathematics, Academic Press, 1968, pp. 423-435.P. D. Rovner, D. A. Henderson: On the implementation of AMBIT/G: a graphical programming language, Proceedings of the 1st international joint conference on Artificial intelligence, ACM, 1969 Both pioneered with data structure diagrams and visual programming as data and patterns were used to be represented by directed-graph diagrams.Brad A. Myers: Taxonomies of visual programming and program visualization, Journal of Visual Languages & Computing, Volume 1, Issue 1, March 1990, pp. 97-123 AMBIT/L was implemented for a PDP-10 computer and used to implement the interactive algebraic manipulation system IAM.

In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time.

In this problem, one must choose time slots for the edges of a wireless communications network so that each node of the network can communicate with each neighboring node without interference. Using a strong edge coloring (and using two time slots for each edge color, one for each direction) would solve the problem but might use more time slots than necessary. Instead, they seek a coloring of the directed graph formed by doubling each undirected edge of the network, with the property that each directed edge has a different color from the edges that go out from and from the neighbors of . They propose a heuristic for this problem based on a distributed algorithm for -edge-coloring together with a postprocessing phase that reschedules edges that might interfere with each other.

Directed graph with arrowheads showing edge directions Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane., p. viii. Node–link diagrams can be traced back to the 14th-16th century works of Pseudo-Lull which were published under the name of Ramon Llull, a 13th century polymath. Pseudo-Lull drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts.. In the case of directed graphs, arrowheads form a commonly used graphical convention to show their orientation; however, user studies have shown that other conventions such as tapering provide this information more effectively.

Time-order between two operations can be represented by an ordered pair of these operations (e.g., the existence of a pair (OP1, OP2) means that OP1 is always before OP2), and a schedule in the general case is a set of such ordered pairs. Such a set, a schedule, is a partial order which can be represented by an acyclic directed graph (or directed acyclic graph, DAG) with operations as nodes and time-order as a directed edge (no cycles are allowed since a cycle means that a first (any) operation on a cycle can be both before and after (any) another second operation on the cycle, which contradicts our perception of Time). In many cases, a graphical representation of such a graph is used to demonstrate a schedule.

Channel routing example Channel routing is the problem of routing of a set of nets N which have fixed terminals on two opposite sides of a rectangle ("channel"). In this context, the horizontal constraint graph is the undirected graph with vertex set N and two nets are connected by an edge if and only if horizontal segments of the routing must overlap. In the given example, only nets 5 and 6 do not have a horizontal constraint between them. The vertical constraint graph is the directed graph with vertex set N and two nets are connected by an edge if and only if there are two pins from different nets on the same vertical line and the edge is directed from the net with pin on the upper edge of the channel.

An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L(G), so the line graph of every Eulerian graph is Hamiltonian. Line graphs may have other Hamiltonian cycles that do not correspond to Euler tours, and in particular the line graph L(G) of every Hamiltonian graph G is itself Hamiltonian, regardless of whether the graph G is Eulerian.. A tournament (with more than two vertices) is Hamiltonian if and only if it is strongly connected. The number of different Hamiltonian cycles in a complete undirected graph on n vertices is and in a complete directed graph on n vertices is .

Since there are m clause components, and the selection of sets of internal edges, L, within each clause component is independent of the selection of sets of internal edges in other clause components, so one can multiply everything to get the weight of Z^M. So, the weight of each Z^M, where M induces a satisfying assignment, is 12^m. Further, where M does not induce a satisfying assignment, M is not proper with respect to some C_j, so the product of the weights of internal edges in Z^M will be 0. The clause component is a weighted, directed graph with 7 nodes with edges weighted and nodes arranged to yield the properties specified above, and is given in Appendix A of Ben-Dor and Halevi (1993).

Generalized to directed graphs, the conjecture has simple counterexamples, as observed by . Here, the chromatic number of a directed graph is just the chromatic number of the underlying graph, but the tensor product has exactly half the number of edges (for directed edges g→g' in G and h→h' in H, the tensor product G × H has only one edge, from (g,h) to (g',h'), while the product of the underlying undirected graphs would have an edge between (g,h') and (g',h) as well). However, the Weak Hedetniemi Conjecture turns out to be equivalent in the directed and undirected settings . The problem cannot be generalized to infinite graphs: gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors.

A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree—or making all its edges point towards the root—in which case it is called an anti-arborescence or in-tree. A rooted tree itself has been defined by some authors as a directed graph. A rooted forest is a disjoint union of rooted trees. A rooted forest may be directed, called a directed rooted forest, either making all its edges point away from the root in each rooted tree—in which case it is called a branching or out-forest—or making all its edges point towards the root in each rooted tree—in which case it is called an anti-branching or in-forest.

However, every planar graph has an arc diagram in which each edge is drawn as a biarc with at most two semicircles. More strongly, every st-planar directed graph (a planar directed acyclic graph with a single source and a single sink, both on the outer face) has an arc diagram in which every edge forms a monotonic curve, with these curves all consistently oriented from one end of the vertex line towards the other. For undirected planar graphs, one way to construct an arc diagram with at most two semicircles per edge is to subdivide the graph and add extra edges so that the resulting graph has a Hamiltonian cycle (and so that each edge is subdivided at most once), and to use the ordering of the vertices on the Hamiltonian cycle as the ordering along the line.

In graph theory, a single-entry single-exit (SESE) region in a given graph is an ordered edge pair (a, b) of distinct control flow edges a and b where: #a dominates b #b postdominates a # Every cycle containing a also contains b and vice versa. where a node x is said to dominate node y in a directed graph if every path from start to y includes x. A node x is said to postdominate a node y if every path from y to end includes x. So, a and b refer to the entry and exit edge, respectively. The first condition ensures that every path from start into the region passes through the region’s entry edge, a. The second condition ensures that every path from inside the region to end passes through the region’s exit edge, b.

Any square matrix A = (a_{ij}) can be viewed as the adjacency matrix of a directed graph, with a_{ij} representing the weight of the edge from vertex i to vertex j. Then, the permanent of A is equal to the sum of the weights of all cycle-covers of the graph; this is a graph-theoretic interpretation of the permanent. #SAT, a function problem related to the Boolean satisfiability problem, is the problem of counting the number of satisfying assignments of a given Boolean formula. It is a #P-complete problem (by definition), as any NP machine can be encoded into a Boolean formula by a process similar to that in Cook's theorem, such that the number of satisfying assignments of the Boolean formula is equal to the number of accepting paths of the NP machine.

Every directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge. The existence of such an ordering can be used to characterize DAGs: a directed graph is a DAG if and only if it has a topological ordering. In general, this ordering is not unique; a DAG has a unique topological ordering if and only if it has a directed path containing all the vertices, in which case the ordering is the same as the order in which the vertices appear in the path.. The family of topological orderings of a DAG is the same as the family of linear extensions of the reachability relation for the DAG,. so any two graphs representing the same partial order have the same set of topological orders.

The graphs that can be built from a single vertex by pendant vertices and true twins, without any false twin operations, are special cases of the Ptolemaic graphs and include the block graphs. The graphs that can be built from a single vertex by false twin and true twin operations, without any pendant vertices, are the cographs, which are therefore distance-hereditary; the cographs are exactly the disjoint unions of diameter-2 distance-hereditary graphs. The neighborhood of any vertex in a distance-hereditary graph is a cograph. The transitive closure of the directed graph formed by choosing any set of orientations for the edges of any tree is distance-hereditary; the special case in which the tree is oriented consistently away from some vertex forms a subclass of distance-hereditary graphs known as the trivially perfect graphs, which are also called chordal cographs..

In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. Skew-symmetric graphs were first introduced under the name of antisymmetrical digraphs by , later as the double covering graphs of polar graphs by , and still later as the double covering graphs of bidirected graphs by . They arise in modeling the search for alternating paths and alternating cycles in algorithms for finding matchings in graphs, in testing whether a still life pattern in Conway's Game of Life may be partitioned into simpler components, in graph drawing, and in the implication graphs used to efficiently solve the 2-satisfiability problem.

As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties: # For every vertex v, σ(v) ≠ v, # For every vertex v, σ(σ(v)) = v, # For every edge (u,v), (σ(v),σ(u)) must also be an edge. One may use the third property to extend σ to an orientation-reversing function on the edges of G. The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism.

Force-directed graph drawing algorithms assign forces among the set of edges and the set of nodes of a graph drawing. Typically, spring-like attractive forces based on Hooke's law are used to attract pairs of endpoints of the graph's edges towards each other, while simultaneously repulsive forces like those of electrically charged particles based on Coulomb's law are used to separate all pairs of nodes. In equilibrium states for this system of forces, the edges tend to have uniform length (because of the spring forces), and nodes that are not connected by an edge tend to be drawn further apart (because of the electrical repulsion). Edge attraction and vertex repulsion forces may be defined using functions that are not based on the physical behavior of springs and particles; for instance, some force-directed systems use springs whose attractive force is logarithmic rather than linear.

Given a purely combinatorial game, there is an associated rooted directed graph whose vertices are game positions and whose edges are moves, and graph traversal starting from the root is used to create a game tree. If the graph contains directed cycles, then a position in the game could repeat infinitely many times, and rules are usually needed to prevent the game from continuing indefinitely. Otherwise, the graph is a directed acyclic graph, and if it isn't a rooted tree, then the game has transpositions. This graph and its topology is important in the study of game complexity, where the state-space complexity is the number of vertices in the graph, the average game length is the average number of vertices traversed from the root to a vertex with no direct successors, and the average branching factor of a game tree is the average outdegree of the graph.

In planning and problem solving, or more formally one-person games, the search space is seen as a directed graph with states as nodes, and transitions as edges. States can have properties, and such a property P is hereditary if for each state S that has P, each state that can be reached from S also has P. The subset of all states that have P plus the subset of all states that have ~P form a partition of the set of states called a hereditary partition. This notion can trivially be extended to more discriminating partitions by instead of properties, considering aspects of states and their domains. If states have an aspect A, with di ⊂ D a partition of the domain D of A, then the subsets of states for which A ∈ di form a hereditary partition of the total set of states iff ∀i, from any state where A ∈ di only other states where A ∈ di can be reached.

Circular layouts are a good fit for communications network topologies such as star or ring networks,. and for the cyclic parts of metabolic networks.. For graphs with a known Hamiltonian cycle, a circular layout allows the cycle to be depicted as the circle, and in this way circular layouts form the basis of the LCF notation for Hamiltonian cubic graphs.. A circular layout may be used on its own for an entire graph drawing, but it also may be used as the layout for smaller clusters of vertices within a larger graph drawing, such as its biconnected components,; . clusters of genes in a gene interaction graph,. or natural subgroups within a social network.. If multiple vertex circles are used in this way, other methods such as force- directed graph drawing may be used to arrange the clusters.. One advantage of a circular layout in some of these applications, such as bioinformatics or social network visualization, is its neutrality:.

Floyd's "tortoise and hare" cycle detection algorithm, applied to the sequence 2, 0, 6, 3, 1, 6, 3, 1, ... Floyd's cycle-finding algorithm is a pointer algorithm that uses only two pointers, which move through the sequence at different speeds. It is also called the "tortoise and the hare algorithm", alluding to Aesop's fable of The Tortoise and the Hare. The algorithm is named after Robert W. Floyd, who was credited with its invention by Donald Knuth.Handbook of Applied Cryptography, by Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, p. 125, describes this algorithm and others However, the algorithm does not appear in Floyd's published work, and this may be a misattribution: Floyd describes algorithms for listing all simple cycles in a directed graph in a 1967 paper, but this paper does not describe the cycle-finding problem in functional graphs that is the subject of this article. In fact, Knuth's statement (in 1969), attributing it to Floyd, without citation, is the first known appearance in print, and it thus may be a folk theorem, not attributable to a single individual.

They are also called Krom formulas, after the work of UC Davis mathematician Melven R. Krom, whose 1967 paper was one of the earliest works on the 2-satisfiability problem.. Each clause in a 2-CNF formula is logically equivalent to an implication from one variable or negated variable to the other. For example, the second clause in the example may be written in any of three equivalent ways: :(x_0\lor\lnot x_3) \;\equiv\; (\lnot x_0\Rightarrow\lnot x_3) \;\equiv\; (x_3\Rightarrow x_0). Because of this equivalence between these different types of operation, a 2-satisfiability instance may also be written in implicative normal form, in which we replace each or clause in the conjunctive normal form by the two implications to which it is equivalent.. A third, more graphical way of describing a 2-satisfiability instance is as an implication graph. An implication graph is a directed graph in which there is one vertex per variable or negated variable, and an edge connecting one vertex to another whenever the corresponding variables are related by an implication in the implicative normal form of the instance.

In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer face of the graph. That is, it is a directed graph drawn without crossings in the plane, in such a way that there are no directed cycles in the graph, exactly one graph vertex has no incoming edges, exactly one graph vertex has no outgoing edges, and these two special vertices both lie on the outer face of the graph.. Within the drawing, each face of the graph must have the same structure: there is one vertex that acts as the source of the face, one vertex that acts as the sink of the face, and all edges within the face are directed along two paths from the source to the sink. If one draws an additional edge from the sink of an st-planar graph back to the source, through the outer face, and then constructs the dual graph (oriented each dual edge clockwise with respect to its primal edge) then the result is again an st-planar graph, augmented with an extra edge in the same way.

A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices.

Formally, a (finite) constraint language (or template) Γ is a finite domain and a finite set of relations over this domain. CSP(Γ) is the constraint satisfaction problem where instances are only allowed to use constraints in Γ. : Theorem (Feder, Vardi 1998): For every constraint language Γ, the problem CSP(Γ) is equivalent under polynomial-time reductions to some H-coloring problem, for some directed graph H. Intuitively, this means that every algorithmic technique or complexity result that applies to H-coloring problems for directed graphs H applies just as well to general CSPs. In particular, one can ask whether the Hell–Nešetřil theorem can be extended to directed graphs. By the above theorem, this is equivalent to the Feder–Vardi conjecture (aka CSP conjecture, dichotomy conjecture) on CSP dichotomy, which states that for every constraint language Γ, CSP(Γ) is NP-complete or in P.. This conjecture was proved in 2017 independently by Dmitry Zhuk and Andrei Bulatov, leading the following corollary: : Corollary (Bulatov 2017; Zhuk 2017): The H-coloring problem on directed graphs, for a fixed H, is either in P or NP-complete.

In his 1736 paper on the Seven Bridges of Königsberg, widely considered to be the birth of graph theory, Leonhard Euler proved that, for a finite undirected graph to have a closed walk that visits each edge exactly once, it is necessary and sufficient that it be connected except for isolated vertices (that is, all edges are contained in one component) and have even degree at each vertex. The corresponding characterization for the existence of a closed walk visiting each edge exactly once in a directed graph is that the graph be strongly connected and have equal numbers of incoming and outgoing edges at each vertex. In either case, the resulting walk is known as an Euler cycle or Euler tour. If a finite undirected graph has even degree at each of its vertices, regardless of whether it is connected, then it is possible to find a set of simple cycles that together cover each edge exactly once: this is Veblen's theorem.. When a connected graph does not meet the conditions of Euler's theorem, a closed walk of minimum length covering each edge at least once can nevertheless be found in polynomial time by solving the route inspection problem.

In theoretical computer science and network routing, Suurballe's algorithm is an algorithm for finding two disjoint paths in a nonnegatively-weighted directed graph, so that both paths connect the same pair of vertices and have minimum total length.. The algorithm was conceived by John W. Suurballe and published in 1974.. The main idea of Suurballe's algorithm is to use Dijkstra's algorithm to find one path, to modify the weights of the graph edges, and then to run Dijkstra's algorithm a second time. The output of the algorithm is formed by combining these two paths, discarding edges that are traversed in opposite directions by the paths, and using the remaining edges to form the two paths to return as the output. The modification to the weights is similar to the weight modification in Johnson's algorithm, and preserves the non-negativity of the weights while allowing the second instance of Dijkstra's algorithm to find the correct second path. The problem of finding two disjoint paths of minimum weight can be seen as a special case of a minimum cost flow problem, where in this case there are two units of "flow" and nodes have unit "capacity".

Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only 2/8 bytes to represent a directed graph, or (by using a packed triangular format and only storing the lower triangular part of the matrix) approximately 2/16 bytes to represent an undirected graph. Although slightly more succinct representations are possible, this method gets close to the information- theoretic lower bound for the minimum number of bits needed to represent all -vertex graphs.. For storing graphs in text files, fewer bits per byte can be used to ensure that all bytes are text characters, for instance by using a Base64 representation.. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a large sparse graph, adjacency lists require less storage space, because they do not waste any space to represent edges that are not present. An alternative form of adjacency matrix (which, however, requires a larger amount of space) replaces the numbers in each element of the matrix with pointers to edge objects (when edges are present) or null pointers (when there is no edge).