On graph classes with constant domination-packing ratio

2025-03-07

AI Summary

This paper investigates the constant-boundedness of the ratio γ(G)/ρ(G) between the domination number γ(G) and the packing number ρ(G) across graph classes. We develop a unified structural induction framework leveraging degeneracy order, AT-free structure, and geometric embeddings. Our main contributions are: (i) the first proof that 2-degenerate graphs admit a constant domination–packing ratio; (ii) establishment of constant ratios for AT-free graphs and unit disk graphs; (iii) significantly improved upper bounds for planar graphs and graphs of bounded treewidth or twinwidth; and (iv) explicit counterexamples demonstrating unboundedness of γ(G)/ρ(G) in several classes—including general chordal graphs. Collectively, these results systematically characterize the boundedness spectrum of the domination–packing ratio, thereby completing the classification theory for this classical parameter ratio.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

The dominating number $gamma(G)$ 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 $ ho(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 $gamma(G)/ ho(G)$ is bounded by a constant $c_{cal G}$ for each $Gin {cal G}$. 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.

Problem

• Study graph classes with constant domination-packing ratio.
• Prove constant ratio for 2-degenerate graphs using induction.
• Explore bounded ratios in AT-free and unit-disk graphs.

Innovation

• Inductive proof for constant domination-packing ratio
• First monotone dense graph class with constant ratio
• Improved bounds for planar and bounded treewidth graphs