A Note on Total and Paired Domination of Cartesian Product Graphs

A dominating set D for a graph G is a subset of V(G) such that any vertex not in D has at least one neighbor in D. The domination number gamma(G) is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs G and H, gamma(G)gamma(H) <= 2 gamma(G square H), and Clark and Suen (2000) proved that gamma(G)gamma(H) <= 2 gamma(G square H). In this paper, we modify the approach of Clark and Suen to prove a similar bounds for total and paired domination in the general case of the n-Cartesian product graph A(1)square ... square A(n). As a by-product of these results, improvements to known total and paired domination inequalities follow as natural corollaries for the standard G square H.