Domination of generalized Cartesian products

The generalized prism π G 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 H that corresponds to the Cartesian product G □ H when π is the identity, and the generalized prism when H is the graph K 2 . Burger, Mynhardt and Weakley [A.P. Burger, C.M. Mynhardt, W.D. 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 ( π G ) = 2 γ ( G ) for any π . In general γ ( G K n ) ≤ n γ ( 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.