Separation of Cartesian Products of Graphs Into Several Connected Components by the Removal of Vertices

A set S subset of V (G) is a vertex k-cut in a graph G = (V (G), E(G)) if G - S has at least k connected components. The k-connectivity of G, denoted as kappa(k)(G), is the minimum cardinality of a vertex k-cut in G. We give several constructions of a set S such that (GH) - S has at least three connected components. Then we prove that for any 2-connected graphs G and H, of order at least six, one of the defined sets S is a minimum vertex 3-cut in GH. This yields a formula for kappa(3)(GH).