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arXiv · 2602.18402

Domination and packing in graphs

Ákos Dúcz·Anna Gujgiczer
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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.

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Ákos Dúcz, Anna Gujgiczer. 2026-02-20. Domination and packing in graphs. https://arxiv.org/abs/2602.18402

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