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arXiv · 2503.04124

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

Shinya Fujita·Boram Park
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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}, $γ_h(G)$, 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 $δ(G)\ge 2$, then $γ_h(G)\le \frac{2n}{5}$, and the bound is tight. We also give some tight upper bounds on $γ_h(G)$ for {triangle-free} graphs $G$ that contain a Hamiltonian path or a Hamiltonian cycle.

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Shinya Fujita, Boram Park. 2025-03-06. Tight upper bounds on the hop domination number of triangle-free graphs. https://arxiv.org/abs/2503.04124

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