biconnected graph

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biconnected graph A graph G, either directed or undirected, with the property that for every three distinct vertices u, v, and w there is a path from u to w not containing v. For an undirected graph, this is equivalent to the graph having no cut vertex.

Two edges of an undirected graph are said to be related either if they are identical or if there is a cycle containing both of them. This is an equivalence relation and partitions the edges into a set of equivalence classes, E1, E2,… En, say. Let Vi be the set of vertices of the edges of Ei for i = 1, 2,… n. Then each graph Gi formed from the vertices Vi and the edges Ei is a biconnected component of G.