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