(Total) Domination in Prisms

With the aid of hypergraph transversals it is proved that $γ_t(Q_{n+1}) = 2γ(Q_n)$, where $γ_t(G)$ and $γ(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the $n$-dimensional hypercube. More generally, it is shown that if $G$ is a bipartite graph, then $γ_t(G \square K_2) = 2γ(G)$. Further, we show that the bipartite condition is essential by constructing, for any $k \ge 1$, a (non-bipartite) graph $G$ such that $γ_t (G \square K_2 ) = 2γ(G) - k$. Along the way several domination-type identities for hypercubes are also obtained.

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