A vertex cut, also called a vertex cut set or separating set (West 2000, p. 148), of a connected graph is a subset of the vertex set such that has more than one connected component. In other words, a vertex cut is a subset of vertices of a connected graph which, if removed (or "cut")--together with any incident edges--disconnects the graph (i.e., forms a disconnected graph).
A vertex cut set of size 1 in a connected graph corresponds to an articulation vertex.
A vertex cut of smallest size in a given connected graph is called a minimum vertex cut and can be found in the Wolfram Language using the function FindVertexCut[G].
The size of a minimum vertex cut in a connected graph gives the vertex connectivity . However, because complete graphs have no vertex cuts (i.e., there is no subset of vertices whose removal disconnected a complete graph), a convention is needed to assign them a vertex connectivity. The convention of letting vertex connectivity for the complete graph allows most general results about connectivity to remain valid on complete graphs (West 2001, p. 149).
The number of vertex cuts in a connected graph with vertex count is related to the number of connected (non-null) vertex-induced subgraphs by
For a not-necessarily-connected graph, a vertex cut of a graph is a vertex set such that has more connected components than (Gross and Yellen 2006, p. 81).