On the Impact of Stability and the Helly Property on the Dominating Set Problem

2026-07-20

AI Summary

This work addresses the Dominating Set problem on graph classes, circumventing the traditional reliance on stability and Helly properties. By refining the progressive exploration framework, the authors devise a fixed-parameter tractable algorithm under the sole assumption that the graph class excludes long co-matchings or double ladders as semi-induced subgraphs. This approach constitutes the first algorithm for Dominating Set that dispenses entirely with stability requirements, achieving a time complexity comparable to existing algorithms for sparse graph classes. The technique extends naturally to Distance-$r$ Dominating Set and Set Cover. The resulting framework applies to broader graph classes—including weakly $\gamma$-closed graphs, nowhere dense graphs, and $K_{t,t}$-free graphs—and the authors further show that relaxing either structural restriction renders the problem W[1]-hard.

Abstract

We extend the algorithmic framework of progressive exploration [Fabiański et al., STACS 2019], which yields simple, yet surprisingly general and efficient parameterized algorithms for Dominating Set, Independent Set, and some of their variants. While they identified stability and the Helly property as necessary for their approach, we show that -- with a simple change -- in the case of Dominating Set, one can get rid of the stability requirement. This yields a fixed-parameter tractable algorithm on exactly those graph classes which do not contain long co-matchings or double-ladders as semi-induced subgraphs. Lifting one of these two restrictions makes Dominating Set W[1]-hard on these classes. Our algorithm generalizes results on weakly $γ$-closed graphs, and results from Sparsity theory, e.g., nowhere dense and biclique-free classes. At the same time, we match the time complexity of the previously known algorithms on those classes. We demonstrate that this technique can easily be applied to the Distance-$r$ Dominating Set and the Set Cover problem.

Problem

• Dominating Set
• stability
• Helly property
• fixed-parameter tractability
• graph classes

Innovation

• Dominating Set
• fixed-parameter tractability
• Helly property
• co-matchings
• semi-induced subgraphs