A unified approach for domination and packing problems in graphs

TL;DR

This paper introduces a unified framework for domination and packing problems in graphs, revealing their equivalence and solving open complexity questions, with polynomial-time solutions for certain graph classes.

Contribution

It generalizes existing concepts into a unified framework, demonstrating their equivalence and providing new polynomial-time algorithms for specific graph classes.

Findings

Equivalence of domination and packing optimization problems.

Polynomial-time solvability for graphs with bounded clique-width.

Polynomial-time solvability for strongly chordal graphs.

Abstract

In this paper, we introduce new concepts of domination and packing functions in graphs, which generalize, respectively, the labelled dominating and packing functions defined by Lee and Chang in 2008, and Hinrichsen et al. in 2019. These generalized functions offer a unified and simpler framework for addressing many of the variations of domination and packing concepts in graphs explored in the literature. Interestingly, their associated optimization problems turn out to be equivalent, providing insight to explain the observed coincidences in computational complexity results for graph classes where both problems, the domination one and its corresponding packing variation, have been analyzed. This equivalence also allows us to solve some computational complexity open questions, for some graph classes. Furthermore, we prove that the generalized problems remain solvable in polynomial time for graphs with bounded clique-width and strongly chordal graphs.