Domination and packing in graphs

TL;DR

This paper investigates the ratio between domination and packing numbers in various graph classes, providing bounds and new results for classes like planar, chordal bipartite, homogeneously orderable graphs, and trees.

Contribution

It establishes improved bounds on the domination-to-packing ratio for several graph classes, extending known results and providing new insights.

Findings

Bounded ratio for planar graphs with an improved upper bound

Established ratio bounds for chordal bipartite graphs

Provided a simple proof for trees

Abstract

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhoods cover all vertices of $G$, while the packing number $\rho(G)$ is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. In this paper we investigate graph classes $\mathcal{G}$ for which the ratio $\gamma(G)/\rho(G)$ is bounded by a constant $c_{\mathcal{G}}$ for every $G \in \mathcal{G}$. Our main result is an improved upper bound on this ratio for planar graphs. We also extend the list of graph classes admitting a bounded ratio by showing this for chordal bipartite graphs and for homogeneously orderable graphs. In addition, we provide a simple, direct proof for trees.