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

TL;DR

The paper studies hop domination in triangle-free graphs, establishing a tight upper bound of $\gamma_h(G) \le \frac{2n}{5}$ for connected triangle-free graphs with $\delta(G)\ge 2$ and $n\ge 15$, with the bound shown to be tight. It develops a framework linking hop domination to the standard domination parameter via the transformation $G^*=Dist(G:2)$ and uses a minimal-counterexample argument with structural analysis around cut-edges and pendant 4-cycles to prove the bound. Additional results provide refined bounds when the graph contains a Hamiltonian path or cycle. It concludes with an open problem regarding hop domination under large girth, proving $f(n,4)=\frac{2n}{5}$ and leaving $r\ge 5$ cases open for future work.

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.

Figures (6)

• Figure 1: The graphs $G_9,G_{14}$ and $G'_{14}$ in $\mathcal{B}$
• Figure 2: A graph $G$ such that $\gamma_h(G)=\frac{2|V(G)|}{5}$
• Figure 3: The graphs $H_1\sim H_7$
• Figure 4: The graphs $C(4,6)$, $C(4,4,6)$, and $C(7,8)$
• Figure 5: Some graphs

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Domination_MS1989
• Remark 1.4
• Proposition 1.5: henning20172
• Theorem 1.6
• Proposition 2.1
• proof
• Theorem 2.2: Domination_path_cycle
• Theorem 2.3