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The game is played on a finite graph with two special nodes, A and B. Each edge of the graph can be either colored or removed. The two players are called Short and Cut, and alternate moves. On Cut's turn, Cut deletes from the graph a non-colored edge of their choice. On Short's turn, Short colors any edge still in the graph.
Each edge in a graph G x may be a virtual edge for at most one SPQR tree edge. An SPQR tree T represents a 2-connected graph G T , formed as follows. Whenever SPQR tree edge xy associates the virtual edge ab of G x with the virtual edge cd of G y , form a single larger graph by merging a and c into a single supervertex, merging b and d into ...
A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction). A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, [5] a directed graph is an ordered pair = (,) comprising:
In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, every bipartite graph can be regarded as the incidence graph of a hypergraph when it is 2-colored and it is indicated which color class corresponds to hypergraph vertices and which to hypergraph edges.
The edge boundary is the set of edges with one endpoint in the inner boundary and one endpoint in the outer boundary. [ 1 ] These boundaries and their sizes are particularly relevant for isoperimetric problems in graphs , separator theorems , minimum cuts , expander graphs , and percolation theory .
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It is straightforward to test whether a graph may be edge colored with one or two colors, so the first nontrivial case of edge coloring is testing whether a graph has a 3-edge-coloring. As Kowalik (2009) showed, it is possible to test whether a graph has a 3-edge-coloring in time O(1.344 n), while using only polynomial space. Although this time ...
Let φ be the automorphism of F(a,b) given by φ(a) = b, φ(b) = ab. Let Γ be the wedge of two loop-edges E a and E b corresponding to the free basis elements a and b, wedged at the vertex v. Let f : Γ → Γ be the map which fixes v and sends the edge E a to E b and that sends the edge E b to the edge-path E a E b.