(Total) Domination in Prisms

Using hypergraph transversals it is proved that γt(Qn+1) = 2γ(Qn), where γt(G) and γ(G) denote the total domination number and the domination number of G, respectively, and Qn is the n-dimensional hypercube. More generally, it is shown that if G is a bipartite graph, then γt(G K2) = 2γ(G). Further, we show that the bipartiteness condition is essential by constructing, for any k > 1, a (non-bipartite) graph G such that γt(G K2) = 2γ(G)− k. Along the way several domination-type identities for hypercubes are also obtained.