On graph classes with constant domination-packing ratio

TL;DR

This work investigates when the domination number $\gamma(G)$ and the packing number $\rho(G)$ of a graph yield a constant upper bound on their ratio across graph classes. It introduces a unified inductive technique based on $(X,Y)$-dominating sets and $(X,Y)$-packings to prove $\gamma(G) \le c_{\mathcal{G}}\rho(G)$ for many classes, including $2$-degenerate, AT-free, convex, unit-disk, planar, treewidth-bounded, and bounded twin-width graphs, with explicit constants. The paper also provides negative results showing unbounded ratios in $3$-degenerate and certain split graphs, and sharpens several constants for classical classes. Overall, it maps the landscape of graph classes with bounded versus unbounded domination-packing ratios and provides a general inductive framework to determine these bounds in future work.

Abstract

The dominating number $γ(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $ρ(G)$ of $G$ is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. In this paper we study graph classes ${\cal G}$ such that $γ(G)/ρ(G)$ is bounded by a constant $c_{\cal G}$ for each $G\in {\cal G}$. We propose an inductive proof technique to prove that if $\cal G$ is the class of $2$-degenerate graphs, then there is such a constant bound $c_{\cal G}$. We note that this is the first monotone, dense graph class that is shown to have constant ratio. We also show that the classes of AT-free and unit-disk graphs have bounded ratio. In addition, our technique gives improved bounds on $c_{\cal G}$ for planar graphs, graphs of bounded treewidth or bounded twin-width. Finally, we provide some new examples of graph classes where the ratio is unbounded.

Figures (8)

• Figure 1: A visual map of graph classes for which the $\gamma/\rho$ ratio has been studied. The classes within rectangular boxes have unbounded ratio, the ones with ellipse boxes have bounded ratio and the dashed border indicates that the problem is not yet settled for the class. The boxes with the grey background highlight the classes for which we established bounds on the ratio and the thick grey borders indicate classes for which we improved the constant bound on the ratio.
• Figure 2: Edge additions in $G^+$.
• Figure 3: The graph $T_2$.
• Figure 4: The embeddings $\pi$ and $\iota$.
• Figure 5: A block with its partition into levels.

Theorems & Definitions (61)

• Conjecture 1.0
• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9