The Counting General Dominating Set Framework

2026-03-15

AI Summary

This study investigates the computational complexity of counting dominating sets and total dominating sets in graphs, with a focus on establishing their #P-completeness in 3-regular planar bipartite simple graphs. To this end, the authors introduce a unified framework—counting generalized dominating sets (#GDS)—which systematically characterizes a broad class of domination-type counting problems for the first time, and develop associated gadget construction techniques within this framework. By modeling the problems as Holant problems and employing reductions via symmetric bipartite graphs together with an extension of Holant dichotomy theorems, they rigorously prove that both counting problems remain #P-complete even under these stringent graph restrictions. This work not only delineates precise complexity boundaries for domination-type counting problems but also offers a novel paradigm for analyzing the complexity of related counting problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Application Domains: Humanities & Computational Social Science

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce a new framework of counting problems called #GDS that encompasses #$(σ, ρ)$-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 propose the technique of gadget construction under 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.

Problem

• counting dominating sets
• counting total dominating sets
• #P-completeness
• 3-regular planar bipartite graphs
• #GDS

Innovation

• #GDS framework
• gadget construction
• Holant dichotomy
• #P-completeness
• dominating set counting