Tight upper bounds on the hop domination number of triangle-free graphs

Abstract

For a graph $G$, a subset $S$ of $V(G)$ is a {\it hop dominating set} of $G$ if every vertex not in $S$ has a $2$-step neighbor in $S$. The {\it hop domination number}, ${\gamma }_h(G)$, of $G$ is the minimum cardinality of a hop dominating set of $G$. In this paper, we show that for a connected triangle-free graph $G$ with $n\ge 15$ vertices, if $\delta (G)\ge 2$, then ${\gamma }_h(G)\le \frac{2n}{5}$, and the bound is tight. We also give some tight upper bounds on ${\gamma }_h(G)$ for {triangle-free} graphs $G$ that contain a Hamiltonian path or a Hamiltonian cycle.