In graph theory, a connected graph *G* is said to be ** k-vertex-connected** (or

**) if it has more than**

*k*-connected*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 *K*_{n}.^{[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.

More generally, the *3-sphere regular cellulation conjecture* claims that every 2-connected graph is the
one-dimensional skeleton of a regular CW-complex on the three-dimensional sphere (http://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf).

## 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 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

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

## Notes

- ^
^{a}^{b}Schrijver,*Combinatorial Optimization*, Springer **^***The algorithm design manual*, p 506, and*Computational discrete mathematics: combinatorics and graph theory with Mathematica*, p. 290-291

## References

- 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, archived from the original on 2012-09-11. - Diestel, Reinhard (2005),
*Graph Theory*(3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-26183-4.