k-vertex-connected graph

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In graph theory, a graph G with vertex set V(G) is said to be k-vertex-connected (or k-connected) for k < |V(G)| if $G \setminus X$ is connected for all $X \subseteq V(G)$ with $\left| X \right| < k$ . In plain English, a graph is k-connected if the graph remains connected when you delete fewer than k vertices from the graph.

In accordance with the definition, the complete graph K_n is (n-1)-connected for n≥2. As a special case that doesn't match the definition, K₁ is regarded as 1-connected.

An equivalent definition for graphs with two or more vertices is that a graph is k-connected if any two of its vertices can be joined by k independent paths. (See Menger's theorem.) (Diestel 2005, p. 55).

A 1-vertex-connected graph is called connected, while a 2-vertex-connected graph is said to be biconnected.

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

Except for the case of K₁, the connectivity of a graph cannot be greater than the minimum degree. This fact is clear for connectivity less than |V(G)|-1 since deleting all neighbors of a vertex of minimum degree will disconnect that vertex from the rest of the graph.

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