Integer domination of Cartesian product graphs

Given a graph G, a dominating set D is a set of vertices such that any vertex not in D has at least one neighbor in D. A {k}-dominating multiset Dk is a multiset of vertices such that any vertex in G has at least k vertices from its closed neighborhood in Dk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) to prove a ‘‘Vizing-like’’ inequality onminimum {k}-dominatingmultisets of graphs G,H and the Cartesian product graph G H . Specifically, denoting the size of a minimum {k}-dominating multiset as γ{k}(G), we demonstrate that γ{k}(G)γ{k}(H) ≤ 2k γ{k}(G H). Published by Elsevier B.V.