Only the g360 symmetry has no degrees of freedom but can seen as directed edges.
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Only the g120 symmetry has no degrees of freedom but can seen as directed edges.
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Only the g96 subgroup has no degrees of freedom but can seen as directed edges.
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These lower symmetries allows degrees of freedoms in defining irregular hexacontatetragons. Only the g64 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g48 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g9 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g11 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g24 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g17 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g19 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g13 subgroup has no degrees of freedom but can seen as directed edges.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as directed edges.
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Mixed graphs are also used as graphical models for Bayesian inference. In this context, an acyclic mixed graph (one with no cycles of directed edges) is also called a chain graph. The directed edges of these graphs are used to indicate a causal connection between two events, in which the outcome of the first event influences the probability of the second event. Undirected edges, instead, indicate a non-causal correlation between two events.
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In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph.
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Financial Networks, the FED and Systemic Risk. Scientific Reports 2. 541. 2 August 2012. . The authors defined financial institutions as nodes and directed edges as lending relations weighted by the amount of outstanding debt.
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A mixed graph G = (V, E, A) is a mathematical object consisting of a set of vertices (or nodes) V, a set of (undirected) edges E, and a set of directed edges (or arcs) A.
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A Loop-carried dependence graph graphically shows the loop-carried dependencies between iterations. Each iteration is listed as a node on the graph, and directed edges show the true, anti, and output dependencies between each iteration.
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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: .
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The result of a WBA is a why–because graph (WBG). The WBG depicts causal relations between factors of an accident. It is a directed acyclic graph where the nodes of the graph are factors. Directed edges denote cause–effect relations between the factors.
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A proof by double counting due to Jim Pitman counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see Double counting (proof technique)#Counting trees.
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The edges are therefore known as directed edges. Example of such network include a link from the reference section on this page which will leads you to another, but not the other way around. In terms of food web, a prey eaten by a predator is another example. :Directed networks can be cyclic or acyclic.
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He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular myriagons. Only the g10000 subgroup has no degrees of freedom but can seen as directed edges.
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He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedoms in defining irregular octacontagons. Only the g80 subgroup has no degrees of freedom but can seen as directed edges.
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He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular hectogons. Only the g100 subgroup has no degrees of freedom but can seen as directed edges.
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A mixed graph is a graph in which some edges may be directed and some may be undirected. It is an ordered triple for a mixed simple graph and for a mixed multigraph with V, E (the undirected edges), A (the directed edges), ϕE and ϕA defined as above. Directed and undirected graphs are special cases.
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Shapes in geometry processing are usually represented as triangle meshes, which can be seen as a graph. Each node in the graph is a vertex (usually in R^3), which has a position. This encodes the geometry of the shape. Directed edges connect these vertices into triangles, which by the right hand rule, then have a direction called the normal.
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Much like Winged Edge, quad-edge structures are used in programs to store the topology of a 2D or 3D polygonal mesh. The mesh itself does not need to be closed in order to form a valid quad-edge structure. Using a quad-edge structure, iterating through the topology is quite easy. Often, the interface to quad- edge topologies is through directed edges.
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A connected component of the undirected subgraph of a chain graph is called a chain. A chain graph may be transformed into an undirected graph by constructing its moral graph, an undirected graph formed from the chain graph by adding undirected edges between pairs of vertices that have outgoing edges to the same chain, and then forgetting the orientations of the directed edges.
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275-278) He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular hexacontagons. Only the g60 symmetry has no degrees of freedom but can seen as directed edges.
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275-278) He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges.
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In the DCM (directed configuration model), each node is given a number of half-edges called tails and heads. Then tails and heads are matched uniformly at random to form directed edges. The size of the giant component, the typical distance, and the diameter of DCM have been studied mathematically. Some real-world complex networks have been modelled by DCM, such as neural networks, finance and social networks.
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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 .
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Much experimental work has been devoted to understanding network motifs in gene regulatory networks. These networks control which genes are expressed in the cell in response to biological signals. The network is defined such that genes are nodes, and directed edges represent the control of one gene by a transcription factor (regulatory protein that binds DNA) encoded by another gene. Thus, network motifs are patterns of genes regulating each other's transcription rate.
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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 }.
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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.
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A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. Some authors restrict the phrase "directed forest" to the case where the edges of each connected component are all directed towards a particular vertex, or all directed away from a particular vertex (see branching).
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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.
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The Symmetries of Things, Chapter 20 r2000000 represents full symmetry and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular megagons. Only the g1000000 subgroup has no degrees of freedom but can be seen as directed edges.
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In a control-flow graph each node in the graph represents a basic block, i.e. a straight-line piece of code without any jumps or jump targets; jump targets start a block, and jumps end a block. Directed edges are used to represent jumps in the control flow. There are, in most presentations, two specially designated blocks: the entry block, through which control enters into the flow graph, and the exit block, through which all control flow leaves.
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In the random surfing model, webgraphs are presented as a sequence of directed graphs G_t,t = 1,2,\ldots such that a graph G_t has t vertices and t edges. The process of defining graphs is parameterized with a probability p, thus we let q= 1-p. Nodes of the model arrive one at time, forming k connections to the existing graph G_t . In some models, connections represent directed edges, and in others, connections represent undirected edges.
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The key difference from the more recent Barabási–Albert model is that the Price model produces a graph with directed edges while the Barabási–Albert model is the same model but with undirected edges. The direction is central to the citation network application which motivated Price. This means that the Price model produces a directed acyclic graph and these networks have distinctive properties. For example, in a directed acyclic graph both longest paths and shortest paths are well defined.
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A polytree (or directed tree or oriented tree or singly connected network) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. Some authors restrict the phrase "directed tree" to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex (see arborescence).
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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.
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This results in a directed edge from T1 to T2 (as T1 has R(A) before T2 having W(A)). # For each case in S where Tj executes a write_item(X) after Ti executes a write_item(X), create an edge (Ti → Tj) in the precedence graph. This results in directed edges from T2 to T1, T2 to T3 and T1 to T3. # The schedule S is serializable if and only if the precedence graph has no cycles.
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Mixed graphs may be used to model job shop scheduling problems in which a collection of tasks is to be performed, subject to certain timing constraints. In this sort of problem, undirected edges may be used to model a constraint that two tasks are incompatible (they cannot be performed simultaneously). Directed edges may be used to model precedence constraints, in which one task must be performed before another. A graph defined in this way from a scheduling problem is called a disjunctive graph.
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In December 1991 the primary architect of the Hurd described the name as a mutually recursive acronym: As both hurd and hird are homophones of the English word herd, the full name GNU Hurd is also a play on the words herd of gnus, reflecting how the kernel works. The logo is called the Hurd boxes and it also reflects on architecture. The logo is a graph where nodes represent the Hurd kernel's servers and directed edges are IPC messages.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g26 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular icosihexagons are d26, an isogonal icosihexagon constructed by thirteen mirrors which can alternate long and short edges, and p26, an isotoxal icosihexagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosihexagon.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g28 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular icosioctagons are d28, an isogonal icosioctagon constructed by ten mirrors which can alternate long and short edges, and p28, an isotoxal icosioctagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosioctagon.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g22 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular icosidigons are d22, an isogonal icosidigon constructed by eleven mirrors which can alternate long and short edges, and p22, an isotoxal icosidigon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosidigon.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g20 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular icosagons are d20, an isogonal icosagon constructed by ten mirrors which can alternate long and short edges, and p20, an isotoxal icosagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosagon.
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The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge. Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then forgets the orientation of the directed edges of the graph.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g10 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular decagons are d10, an isogonal decagon constructed by five mirrors which can alternate long and short edges, and p10, an isotoxal decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular decagon.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g14 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular tetradecagons are d14, an isogonal tetradecagon constructed by seven mirrors which can alternate long and short edges, and p14, an isotoxal tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular tetradecagon.
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g34 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular triacontatetragons are d34, an isogonal triacontatetragon constructed by seventeen mirrors which can alternate long and short edges, and p34, an isotoxal triacontatetragon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular triacontatetragon.
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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 .
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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.
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Then, we add the segments from the subdivision, one by one, in random order, refining the trapezoidal decomposition. Using backwards analysis, we can show that the expected number of trapezoids created for each insertion is bounded by a constant. We build a directed acyclic graph, where the vertices are the trapezoids that existed at some point in the refinement, and the directed edges connect the trapezoids obtained by subdivision. The expected depth of a search in this digraph, starting from the vertex corresponding to the bounding box, is O(log n).
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For two convex polygons and in the plane with and vertices, their Minkowski sum is a convex polygon with at most + vertices and may be computed in time O ( + ) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by polar angle. Let us merge the ordered sequences of the directed edges from and into a single ordered sequence .
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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.
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GQL graphs can be mixed: they can contain directed edges, where one of the endpoint nodes of an edge is the tail (or source) and the other node is the head (or target or destination), but they can also contain undirected (bidirectional or reflexive) edges. Nodes and edges, collectively known as elements, have attributes. Those attributes may be data values, or labels (tags). Values of properties cannot be elements of graphs, nor can they be whole graphs: these restrictions intentionally force a clean separation between the topology of a graph, and the attributes carrying data values in the context of a graph topology.
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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.
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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.
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A sparsest cut of a graph G=(V,E) is a partition for which the ratio of the number of edges connecting the two partitioned components over the product of the numbers of nodes of both components is minimized. This is a NP-hard problem, and it can be approximated to within O(\log n) factor using Theorem 2. Also, a sparsest cut problem with weighted edges, weighted nodes or directed edges can be approximated within an O(\log p) factor where is the number of nodes with nonzero weight according to Theorem 3, 4 and 5.
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The inner plate represents the variables associated with each of the N_i words in document i: z_{ij} is the topic distribution for the jth word in document i, and w_{ij} is the actual word used. The N in the corner represents the repetition of the variables in the inner plate N_i times, once for each word in document i. The circle representing the individual words is shaded, indicating that each w_{ij} is observable, and the other circles are empty, indicating that the other variables are latent variables. The directed edges between variables indicate dependencies between the variables: for example, each w_{ij} depends on z_{ij} and β.
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Every Grothendieck topos is an elementary topos, but the converse is not true (since every Grothendieck topos is cocomplete, which is not required from an elementary topos). The categories of finite sets, of finite G-sets (actions of a group G on a finite set), and of finite graphs are elementary topoi that are not Grothendieck topoi. If C is a small category, then the functor category SetC (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos.
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The elements of IBIS are: issues (questions that need to be answered), each of which are associated with (answered by) alternative positions (possible answers or ideas), which are associated with arguments which support or object to a given position; arguments that support a position are called "pros", and arguments that object to a position are called "cons". In the course of the treatment of issues, new issues come up which are treated likewise. IBIS elements are usually represented as nodes, and the associations between elements are represented as directed edges (arrows). IBIS as typically used has three basic elements (or kinds of nodes, labeled "issue", "position", "argument") and a limited set of ways that the nodes may be connected.
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The Bondy–Chvátal theorem states that a graph is Hamiltonian if and only if its closure is Hamiltonian; since the complete graph is Hamiltonian, Ore's theorem is an immediate consequence. found a version of Ore's theorem that applies to directed graphs. Suppose a digraph G has the property that, for every two vertices u and v, either there is an edge from u to v or the outdegree of u plus the indegree of v equals or exceeds the number of vertices in G. Then, according to Woodall's theorem, G contains a directed Hamiltonian cycle. Ore's theorem may be obtained from Woodall by replacing every edge in a given undirected graph by a pair of directed edges.
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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.
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In this case each matrix can be encoded as a directed edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges. The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from A to B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem.
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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.
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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.
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And a pseudoforest may be determined by specifying, for each of its nodes, the endpoint of the edge extending outwards from that node; there are n possible choices for the endpoint of a single edge (allowing self-loops) and therefore nn possible pseudoforests. By finding a bijection between trees with two labeled nodes and pseudoforests, Joyal's proof shows that Tn = nn − 2. Finally, the fourth proof of Cayley's formula presented by Aigner and Ziegler is a double counting proof due to Jim Pitman. In this proof, Pitman considers the sequences of directed edges that may be added to an n-node empty graph to form from it a single rooted tree, and counts the number of such sequences in two different ways.
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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.
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In information visualization, Van Wijk is known for his research in texture synthesis, treemaps,.. and flow visualization... His work on map projection. won the 2009 Henry Johns Award of the British Cartographic Society for best cartographic journal article... He has twice been program co-chair for IEEE Visualization, and once for IEEE InfoVis. In 2007, he received an IEEE Technical Achievement Award for his visualization research.. In graph drawing, van Wijk has worked on the visualization of small-world networks. and on the depiction of abstract trees as biological trees.. He has also conducted user studies that showed that the standard depiction of directed edges in graph drawings using arrowheads is less effective at conveying the directionality of the edges to readers than other conventions such as tapering... He was one of two invited speakers at the 19th International Symposium on Graph Drawing in 2011,Graph Drawing 2011 program, retrieved 2011-11-10.
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The problem of planning a freight delivery system may be modeled by a network in which the vertices represent cities and the (undirected) edges represent potential freight delivery routes between pairs of cities. Each route can achieve a certain profit, but can only be used if freight depots are constructed at both its ends, with a certain cost. The problem of designing a network that maximizes the difference between the profits and the costs can be solved as a closure problem, by subdividing each undirected edge into two directed edges, both directed outwards from the subdivision point. The weight of each subdivision point is a positive number, the profit of the corresponding route, and the weight of each original graph vertex is a negative number, the cost of building a depot in that city.. Together with open pit mining, this was one of the original motivating applications for studying the closure problem; it was originally studied in 1970, in two independent papers published in the same issue of the same journal by J. M. W. Rhys and Michel Balinski...
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A common way to form covering graphs uses voltage graphs, in which the darts of the given graph G (that is, pairs of directed edges corresponding to the undirected edges of G) are labeled with inverse pairs of elements from some group. The derived graph of the voltage graph has as its vertices the pairs (v,x) where v is a vertex of G and x is a group element; a dart from v to w labeled with the group element y in G corresponds to an edge from (v,x) to (w,xy) in the derived graph. The universal cover can be seen in this way as a derived graph of a voltage graph in which the edges of a spanning tree of the graph are labeled by the identity element of the group, and each remaining pair of darts is labeled by a distinct generating element of a free group. The bipartite double can be seen in this way as a derived graph of a voltage graph in which each dart is labeled by the nonzero element of the group of order two.
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