One may also consider playing either Geography game on an undirected graph (that is, the edges can be traversed in both directions). Fraenkel, Scheinerman, and Ullman show that undirected vertex geography can be solved in polynomial time, whereas undirected edge geography is PSPACE- complete, even for planar graphs with maximum degree 3. If the graph is bipartite, then Undirected Edge Geography is solvable in polynomial time.
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In the undirected edge- disjoint paths problem, we are given an undirected graph and two vertices and , and we have to find the maximum number of edge-disjoint s-t paths in . The Menger's theorem states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
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The Rado graph: for instance there is an edge from 0 to 3 because the 0th bit of 3 is non zero. Ackermann in 1937 and Richard Rado in 1964 used this predicate to construct the infinite Rado graph. In their construction, the vertices of this graph correspond to the non-negative integers, written in binary, and there is an undirected edge from vertex i to vertex j, for i < j, when BIT(j,i) is nonzero..
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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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In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties. Given a vertex v and an edge label i, the rotation map returns the i'th neighbor of v and the edge label that would lead back to v.
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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.
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The Kleinberg model of a network is effective at demonstrating the effectiveness of greedy small world routing. The model uses an n x n grid of nodes to represent a network, where each node is connected with an undirected edge to its neighbors. To give it the “small world” effect, a number of long range edges are added to the network that tend to favor nodes closer in distance rather than farther. When adding edges, the probability of connecting some random vertex v to another random vertex w is proportional to 1/d(v,w)^q, where q is the clustering exponent.
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Kruskal's algorithm finds a minimum spanning forest of an undirected edge- weighted graph. If the graph is connected, it finds a minimum spanning tree. (A minimum spanning tree of a connected graph is a subset of the edges that forms a tree that includes every vertex, where the sum of the weights of all the edges in the tree is minimized. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.) It is a greedy algorithm in graph theory as in each step it adds the next lowest-weight edge that will not form a cycle to the minimum spanning forest.
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An unrooted binary tree T may be transformed into a full rooted binary tree (that is, a rooted tree in which each non-leaf node has exactly two children) by choosing a root edge e of T, placing a new root node in the middle of e, and directing every edge of the resulting subdivided tree away from the root node. Conversely, any full rooted binary tree may be transformed into an unrooted binary tree by removing the root node, replacing the path between its two children by a single undirected edge, and suppressing the orientation of the remaining edges in the graph. For this reason, there are exactly 2n −3 times as many full rooted binary trees with n leaves as there are unrooted binary trees with n leaves.
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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.
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If e is in the first subset of edges at v, these two edges are from u0 into v0 and from v1 into u1, while if e is in the second subset, the edges are from u0 into v1 and from v0 into u1. In the other direction, given a skew- symmetric graph G, one may form a polar graph that has one vertex for every corresponding pair of vertices in G and one undirected edge for every corresponding pair of edges in G. The undirected edges at each vertex of the polar graph may be partitioned into two subsets according to which vertex of the polar graph they go out of and come into. A regular path or cycle of a skew-symmetric graph corresponds to a path or cycle in the polar graph that uses at most one edge from each subset of edges at each of its vertices.
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Any partial order may be represented (usually in more than one way) by a directed acyclic graph in which there is a path from x to y whenever x and y are elements of the partial order with . The graphs that represent series-parallel partial orders in this way have been called vertex series parallel graphs, and their transitive reductions (the graphs of the covering relations of the partial order) are called minimal vertex series parallel graphs. Directed trees and (two- terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallel partial orders may be used to represent reachability relations in directed trees and series parallel graphs. The comparability graph of a partial order is the undirected graph with a vertex for each element and an undirected edge for each pair of distinct elements x, y with either or .
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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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