Domination and packing in graphs
Ákos Dúcz, Anna Gujgiczer
TL;DR
The work studies the domination–packing gap in graphs by asking whether a constant $c$ exists with $\gamma(G) \le c\,\rho(G)$ for graph classes, focusing on planar graphs, chordal bipartite graphs, homogeneously orderable graphs, and trees. It develops a planar-specific framework combining a strengthened induction with $X$-domination/$X$-packing and a discharging argument to establish $\gamma(G) \le 7\rho(G)$ for planar graphs, and it proves tight constants $\gamma(G) \le 2\rho(G)$ for both chordal bipartite and homogeneously orderable graphs. A simple, direct proof is also provided for trees, showing $\gamma(T)=\rho(T)$. Collectively, these results advance understanding of when domination can be bounded by packing in sparse graph classes, and they raise open questions about the optimal planar bound and extensions to other geometric graph families, with potential practical implications for network monitoring and resource placement problems.
Abstract
The dominating number $γ(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhoods cover all vertices of $G$, while the packing number $ρ(G)$ 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 $γ(G)/ρ(G)$ 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.