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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 .
If G is a graph, the line graph L(G) has a vertex for each edge of G, and an edge for each pair of adjacent edges in G. Thus, the chromatic number of L(G) equals the chromatic index of G. If G is bipartite, the cliques in L(G) are exactly the sets of edges in G sharing a common endpoint. Now Kőnig's line coloring theorem, stating that the ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
Then one endpoint of edge e is in set V and the other is not. Since tree Y 1 is a spanning tree of graph P, there is a path in tree Y 1 joining the two endpoints. As one travels along the path, one must encounter an edge f joining a vertex in set V to one that is not in set V.
The edge-connectivity version of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two distinct vertices. Then the size of the minimum edge cut for x and y (the minimum number of edges whose removal disconnects x and y) is equal to the maximum number of pairwise edge-disjoint paths from x to y.
The maximum common edge subgraph problem on general graphs is NP-complete as it is a generalization of subgraph isomorphism: a graph is isomorphic to a subgraph of another graph if and only if the maximum common edge subgraph of and has the same number of edges as .
If s and t are specified vertices of the graph G, then an s – t cut is a cut in which s belongs to the set S and t belongs to the set T. In an unweighted undirected graph, the size or weight of a cut is the number of edges crossing the cut. In a weighted graph, the value or weight is defined by the sum of the weights of the edges crossing the ...
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 discrete 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 ...