# 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.

Relevant topics on |

Graph connectivity |
---|

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*.

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 |

## Examples

## See also

## References

- Eric W. Weisstein. "Biconnected Graph." From MathWorldâ€”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).