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 (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, γ(G) γ(H) ≤ (G{ballot box}H), and Clark and Suen (2000) proved that γ(G)γ (H) ≤ γ (G{ballot box}H). In this paper, we modify the approach of Clark and Suen to prove similar bounds for total and paired domination in the general case of the n-Cartesian product graph A₁{ballot box}...{ballot box}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{ballot box}H.