Counting dominating sets and related structures in graphs

TL;DR

This paper investigates the maximum number of dominating sets in regular graphs and extends entropy-based methods to related enumeration problems, including vertex colorings and existence homomorphisms.

Contribution

It introduces entropy techniques to bound and analyze the count of dominating sets and related structures, broadening their application to various graph parameters.

Findings

Entropy methods effectively bound dominating set counts

Extensions to vertex coloring enumeration problems

Partial results on existence homomorphisms

Abstract

We consider some problems concerning the maximum number of (strong) dominating sets in a regular graph, and their weighted analogues. Our primary tool is Shearer's entropy lemma. These techniques extend to a reasonably broad class of graph parameters enumerating vertex colorings satisfying conditions on the multiset of colors appearing in (closed) neighborhoods. We also generalize further to enumeration problems for what we call existence homomorphisms. Here our results are substantially less complete, though we do solve some natural problems.