A graph with connectivity 4.

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.

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] Clearly the complete graph with n vertices has connectivity n − 1 under this definition.

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

Applications

Components

Every graph decomposes into a tree of 1-connected components. 1-connected graphs decompose into a tree of biconnected components. 2-connected graphs decompose into a tree of triconnected components.

Polyhedral combinatorics

The 1-skeleton of any k-dimensional convex polytope forms a k-vertex-connected graph (Balinski's theorem).[2] 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[3] consider all possible pairs of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for 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 and with capacity 1 to each edge, noting that a flow of in this graph corresponds, by the integral flow theorem, to pairwise edge-independent paths from to .

See also

Notes

  1. 1 2 Schrijver (12 February 2003), Combinatorial Optimization, Springer, ISBN 9783540443896
  2. ↑ Balinski, M. L. (1961), "On the graph structure of convex polyhedra in n-space", Pacific Journal of Mathematics, 11 (2): 431–434, doi:10.2140/pjm.1961.11.431.
  3. ↑ The algorithm design manual, p 506, and Computational discrete mathematics: combinatorics and graph theory with Mathematica, p. 290-291

References

  • Diestel, Reinhard (2005), Graph Theory (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-26183-4.
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