Domination and packing in graphs

TL;DR

This paper investigates relationships between domination and packing numbers in various classes of graphs, providing new bounds and confirming tightness for biconvex graphs.

Contribution

It establishes new bounds on domination numbers relative to packing numbers for bipartite cubic, maximal outerplanar, and biconvex graphs, advancing understanding of these parameters.

Findings

Proves $ ho(G) \\leq \\gamma(G)$ for all graphs.

Shows $\\gamma(G) \\leq rac{120}{49} \\rho(G)$ for bipartite cubic graphs.

Demonstrates tight bound $\\gamma(G) \\leq 2 \\rho(G)$ for biconvex graphs.

Abstract

Given a graph~$G$, the domination number, denoted by~$\gamma(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$\rho(G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$\rho(G) \leq \gamma(G)$ is well-known. Henning et al.\ conjectured that~$\gamma(G) \leq 2\rho(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${\gamma(G) \leq \frac{120}{49}\rho(G)}$ if~$G$ is a bipartite cubic graph. We also show that $\gamma(G) \leq 3\rho(G)$ if~$G$ is a maximal outerplanar graph, and that~$\gamma(G) \leq 2\rho(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight.