Domination of generalized Cartesian products

The generalized prism πG of G is the graph consisting of two copies of G, with edges between the copies determined by a permutation π acting on the vertices of G. We define a generalized Cartesian product G π H that corresponds to the Cartesian product G H when π is the identity, and the generalized prism when H is the graph K2. Burger, Mynhardt and Weakley [On the domination number of prisms of graphs, Discuss. Math. Graph Theory 24(2) (2004), 303–318] characterized universal doublers, i.e. graphs for which γ(πG) = 2γ(G) for any π. In general γ(G π Kn) ≤ nγ(G) for any n ≥ 2 and permutation π, and a graph attaining equality in this upper bound for all π is called a universal multiplier. We characterize such graphs and consider a similar problem for the product G π Cn.