Biconnected graph
{{Short description|Type of graph}}
{{Graph connectivity sidebar}}
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
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.
| class="wikitable" style="text-align:center"
|+Nonseparable (or 2-connected) graphs (or blocks) with n nodes {{OEIS|id=A002218}} | |
| 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
See also
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
- {{citation
| last = Diestel | first = Reinhard | author-link = Reinhard Diestel
| edition = 5th
| isbn = 978-3-662-53621-6
| location = Berlin, New York
| publisher = Springer-Verlag
| title = Graph Theory
| url = https://diestel-graph-theory.com/index.html
| year = 2016}}.
External links
- [https://code.google.com/p/jbpt/ The tree of the biconnected components Java implementation] in the jBPT library (see BCTree class).