Separation of Cartesian products of graphs into several connected components by the removal of vertices

A set S ⊆ 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 κk(G), is the minimum cardinality of a vertex k-cut in G. We give several constructions of a set S such that (G2H)− 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 G2H. This yields a formula for κ3(G2H).