Biconnected graph

In graph theory, a biconnected graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain connected. Therefore a biconnected graph has no articulation vertices. The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not regarded as 2-connected. This property is especially useful in maintaining a graph with a two-fold redundancy, to prevent disconnection upon the removal of a single edge (or connection). The use of biconnected graphs is very important in the field of networking (see Network flow), because of this property of redundancy. A biconnected undirected graph is a connected graph that is not broken into disconnected pieces by deleting any single vertex (and its incident edges). A biconnected directed graph is one such that for any two vertices v and w there are two directed paths from v to w which have no vertices in common other than v and w. File:4 Node Biconnected.svg|A biconnected graph on four vertices and four edges File:4 Node Not-Biconnected.svg|A graph that is not biconnected. The removal of vertex x would disconnect the graph. File:5 Node Biconnected.svg|A biconnected graph on five vertices and six edges File:5 Node Not-Biconnected.svg|A graph that is not biconnected. The removal of vertex x would disconnect the graph.