# k-vertex-connected graph

Short description: Graph which remains connected when k or fewer nodes removed
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.[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 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 $\displaystyle{ (s, t) }$ of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for $\displaystyle{ (s, t) }$ 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 $\displaystyle{ s }$ and $\displaystyle{ t }$ with capacity 1 to each edge, noting that a flow of $\displaystyle{ k }$ in this graph corresponds, by the integral flow theorem, to $\displaystyle{ k }$ pairwise edge-independent paths from $\displaystyle{ s }$ to $\displaystyle{ t }$.