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.

Definition 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.

Examples

+Nonseparable (or 2-connected) graphs (or blocks) with n nodes Vertices Number of Possibilities 1 0 2 1 3 1 4 3 5 10 6 56 7 468 8 7123 9 194066 10 9743542 11 900969091 12 153620333545 13 48432939150704 14 28361824488394169 15 30995890806033380784 16 63501635429109597504951 17 244852079292073376010411280 18 1783160594069429925952824734641 19 24603887051350945867492816663958981

Structure of 2-connected graphs Every 2-connected graph can be constructed inductively by adding paths to a cycle .

See also Biconnected component

References Eric W. Weisstein. "Biconnected Graph." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BiconnectedGraph.html Paul E. Black, "biconnected graph", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. (accessed TODAY) Available from: https://xlinux.nist.gov/dads/HTML/biconnectedGraph.html .

External links The tree of the biconnected components Java implementation in the jBPT library (see BCTree class).