Improved Domination--Packing Bounds in Claw-Free Cubic Graphs and Unit Disk Graphs

Abstract

Given a graph $G$, the domination number $γ(G)$ is the minimum cardinality of a dominating set in $G$, and the packing number $ρ(G)$ is the maximum cardinality of a set of vertices that are pairwise at distance at least $3$. The ratio between these parameters has been widely studied in several graph classes. It is known that $γ(G) \le 2ρ(G)$ for claw-free subcubic graphs, up to finitely many exceptions, and that $γ(G) \le 32ρ(G)$ for unit disk graphs. In this paper, we improve the latter bound by showing that $γ(G) \le 16ρ(G)$ for a unit disk graph $G$. For the former bound, we show that it can be improved in the cubic bridgeless setting; more precisely, every bridgeless claw-free cubic graph $G$ satisfies $γ(G) \le \frac{7}{4}ρ(G) + \frac{5}{6}$. These results are not tight. In fact, we give example of an infinite family of bridgeless cubic graphs $G$ with $γ(G) = 5ρ(G)/4$ and an infnite family of unit disk graphs $G$ in which $γ(G) = 3ρ(G)$.