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

2026-06-28

AI Summary

This study investigates the upper bounds on the domination-packing ratio, γ(G)/ρ(G), in claw-free cubic graphs and unit disk graphs. By employing combinatorial graph theory, structural analysis, and constructive proofs, the authors improve the known upper bound for unit disk graphs from 32 to 16, establishing that γ(G) ≤ 16ρ(G). For bridgeless claw-free cubic graphs, they prove a tighter bound: γ(G) ≤ (7/4)ρ(G) + 5/6. Furthermore, the paper constructs two infinite families of graphs achieving γ(G) = (5/4)ρ(G) and γ(G) = 3ρ(G), respectively, thereby demonstrating the tightness of the lower bounds and significantly advancing the current theoretical understanding of domination and packing parameters in these graph classes.

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 $γ(G) = 3ρ(G)$.

Problem

• domination number
• packing number
• claw-free cubic graphs
• unit disk graphs
• graph domination

Innovation

• domination number
• packing number
• claw-free cubic graphs
• unit disk graphs
• graph domination bounds