k-vertex-connected graph

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed.

The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.

Definitions

A graph (other than a complete graph) has connectivity k if k is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them.[1] Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with n vertices has connectivity n  1, as implied by the first definition.

An equivalent definition is that a graph with at least two vertices is k-connected if, for every pair of its vertices, it is possible to find k vertex-independent paths connecting these vertices; see Menger's theorem (Diestel 2005, p. 55). This definition produces the same answer, n  1, for the connectivity of the complete graph Kn.[1]

A 1-connected graph is called connected; a 2-connected graph is called biconnected. A 3-connected graph is called triconnected.

Applications

Polyhedral Combinatorics

The 1-skeleton of any k-dimensional convex polytope forms a k-vertex-connected graph (Balinski's theorem, Balinski 1961). As a partial converse, Steinitz's theorem states that any 3-vertex-connected planar graph forms the skeleton of a convex polyhedron.

Computational complexity

The vertex-connectivity of an input graph G can be computed in polynomial time in the following way[2] consider all possible pairs (s, t) of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for (s, t) is the number of pairwise vertex-independent paths between them, encode the input by doubling each vertex as an edge to reduce to a computation of the number of pairwise edge-independent paths, and compute the maximum number of such paths by computing the maximum flow in the graph between s and t with capacity 1 to each edge, noting that a flow of k in this graph corresponds, by the integral flow theorem, to k pairwise edge-independent paths from s to t.

See also

Notes

  1. 1 2 Schrijver, Combinatorial Optimization, Springer
  2. The algorithm design manual, p 506, and Computational discrete mathematics: combinatorics and graph theory with Mathematica, p. 290-291

References

This article is issued from Wikipedia - version of the Tuesday, March 17, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.