The Counting General Dominating Set Framework

TL;DR

This paper introduces the #GDS framework for counting domination problems, demonstrating its relation to Holant and proving #P-completeness for counting dominating and total dominating sets in specific graph classes.

Contribution

It develops a new counting framework #GDS, adapts Holant gadget techniques, and establishes #P-completeness results for key domination counting problems.

Findings

Proves #P-completeness of counting dominating sets in 3-regular planar bipartite graphs.

Establishes #P-completeness of counting total dominating sets in the same graph class.

Connects #GDS framework with Holant problems and extends dichotomy results.

Abstract

We introduce a new framework of counting problems called #GDS that encompasses #$(\sigma, \rho)$-Set, a class of domination-type problems that includes counting dominating sets and counting total dominating sets. We explore the intricate relation between #GDS and the well-known Holant. We adapt the technique of gadget construction of Holant to the #GDS framework; using this technique, we prove the #P-completeness of counting dominating sets for 3-regular planar bipartite simple graphs. Through a generalization of a Holant dichotomy, and a special reduction method via symmetric bipartite graphs, we also prove the #P-completeness of counting total dominating sets for the same graph class.