For k ≥ 2, a graph G is said to be k-vertex connected, or simply k-connected, when |V(G)| ≥ k + 1 and the removal of any k − 1 vertices (and any incident edges) from G does not result in a disconnected graph. We use 1-connected as a synonym for connected.

If a graph G with at least k + 1 vertices is not k-connected and the deletion of a set S of k − 1 vertices disconnects it, there is a partition of V(G) \ S into nonempty sets X, Y with no edges crossing (one end in X, one in Y). Let H and K be the subgraphs induced by X ∪ S and Y ∪ S, except that edges with both ends in S are to be put in one and only one of H or K. Then we obtain edge-disjoint subgraphs H and K whose union is G and such that |V(H)∩V(K)| = k − 1. Conversely, if such subgraphs H and K exist and each contains at least one more vertex than their intersection, then G is not k-connected.

A graph is said to be nonseparable when it is 2-connected and has no loops, or when it is a bond-graph (with two vertices and any positive number of edges joining them, including the link-graph with one such edge), a loop-graph (one edge joining a single vertex to itself), or a vertex-graph. All polygons, for example, are nonseparable; path-graphs or other trees with at least two edges are not.