On the vertices belonging to all, some, none minimum dominating set

TL;DR

This paper characterizes vertices in relation to all, some, or none of the minimum dominating sets across various graph classes, providing insights and answering open questions in the field.

Contribution

It offers a detailed characterization of vertices in minimum dominating sets and addresses open problems for specific graph subclasses.

Findings

Vertices are classified based on their membership in minimum dominating sets.

Characterizations are refined for chordal, claw-free, and triangle-free graphs.

The paper provides counterexamples and solutions to open questions.

Abstract

We characterize the vertices belonging to all minimum dominating sets, to some minimum dominating sets but not all, and to no minimum dominating set. We refine this characterization for some well studied sub-classes of graphs: chordal, claw-free, triangle-free. Also we exhibit some graphs answering to some open questions of the literature on minimum dominating sets.