Characterizing Cartesian fixers and multipliers

Let G H denote the Cartesian product of the graphs G and H. In 2004, Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24(3) (2004), 389–402] characterized prism fixers, i.e., graphs G for which γ(G K2) = γ(G), and noted that γ(G Kn) ≥ min{|V (G)|, γ(G)+n−2}. We call a graph G a consistent fixer if γ(G Kn) = γ(G)+n−2 for each n such that 2 ≤ n < |V (G)|−γ(G)+2, and characterize this class of graphs. Also in 2004, Burger, Mynhardt and Weakley [On the domination number of prisms of graphs, Dicuss. Math. Graph Theory 24(2) (2004), 303–318] characterized prism doublers, i.e., graphs G for which γ(G K2) = 2γ(G). In general γ(G Kn) ≤ nγ(G) for any n ≥ 2. We call a graph attaining equality in this bound a Cartesian n-multiplier and also characterize this class of graphs.