*k*-vertex-connected graph

__: Graph which remains connected when k or fewer nodes removed__

**Short description**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.^{[citation needed]}

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). This definition produces the same answer, *n* − 1, for the connectivity of the complete graph *K*_{n}.^{[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 [math]\displaystyle{ (s, t) }[/math] of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for [math]\displaystyle{ (s, t) }[/math] 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 [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math] with capacity 1 to each edge, noting that a flow of [math]\displaystyle{ k }[/math] in this graph corresponds, by the integral flow theorem, to [math]\displaystyle{ k }[/math] pairwise edge-independent paths from [math]\displaystyle{ s }[/math] to [math]\displaystyle{ t }[/math].

## See also

*k*-edge-connected graph- Connectivity (graph theory)
- Menger's theorem
- Structural cohesion
- Tutte embedding
- Vertex separator

## Notes

- ↑
^{1.0}^{1.1}Schrijver (12 February 2003),*Combinatorial Optimization*, Springer, ISBN 9783540443896, https://books.google.com/books?id=mqGeSQ6dJycC&q=%22k-vertex-connected+%22 - ↑ 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. - ↑
*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, http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/.

Original source: https://en.wikipedia.org/wiki/K-vertex-connected graph.
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