Geodesic
(Total) domination in prisms
The electronic journal of combinatorics, Tome 24 (2017) no. 1
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Using 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 bipartiteness 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.
DOI : 10.37236/6288
Classification : 05C69, 05C76
Mots-clés : domination, total domination, hypercube, Cartesian product of graphs, covering codes, hypergraph transversal 
@article{10_37236_6288,
     author = {Jernej Azarija and Michael Henning and Sandi Klav\v{z}ar},
     title = {(Total) domination in prisms},
     journal = {The electronic journal of combinatorics},
     year = {2017},
     volume = {24},
     number = {1},
     doi = {10.37236/6288},
     zbl = {1355.05181},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/6288/}
}
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Jernej Azarija; Michael Henning; Sandi Klavžar. (Total) domination in prisms. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/6288

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