(Total) Domination in Prisms

TL;DR

This paper proves a relationship between total domination and domination numbers in hypercubes and bipartite graphs, introduces new identities, and shows the necessity of bipartiteness with counterexamples.

Contribution

It establishes a formula for total domination in hypercubes and bipartite graphs, and demonstrates the importance of bipartiteness with constructed counterexamples.

Findings

entity: entity: entity for hypercubes

entity: entity for bipartite graphs

Counterexamples for non-bipartite graphs

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.