Integer domination of Cartesian product graphs

Given a graph GG, a dominating set DD is a set of vertices such that any vertex not in DD has at least one neighbor in DD. A {k}{k}-dominating multiset DkDk is a multiset of vertices such that any vertex in GG has at least kk vertices from its closed neighborhood in DkDk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) to prove a “Vizing-like” inequality on minimum {k}{k}-dominating multisets of graphs G,HG,H and the Cartesian product graph G□HG□H. Specifically, denoting the size of a minimum {k}{k}-dominating multiset as γ{k}(G)γ{k}(G), we demonstrate that γ{k}(G)γ{k}(H)≤2kγ{k}(G□H).