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arXiv · 1606.08143

(Total) Domination in Prisms

Jernej Azarija·Michael A. Henning·Sandi Klavžar
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Abstract

With the aid of hypergraph transversals it is proved that $\gamma_t(Q_{n+1}) = 2\gamma(Q_n)$, where $\gamma_t(G)$ and $\gamma(G)$ 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 $\gamma_t(G \square K_2) = 2\gamma(G)$. Further, we show that the bipartite condition is essential by constructing, for any $k \ge 1$, a (non-bipartite) graph $G$ such that $\gamma_t (G \square K_2 ) = 2\gamma(G) - k$. Along the way several domination-type identities for hypercubes are also obtained.

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Jernej Azarija, Michael A. Henning, Sandi Klavžar. 2016-06-27. (Total) Domination in Prisms. https://arxiv.org/abs/1606.08143

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