The Counting General Dominating Set Framework
Computational Complexity
2026-05-20 v2
Abstract
We introduce a new framework of counting problems called #GDS that encompasses #-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.
Keywords
Cite
@article{arxiv.2603.14749,
title = {The Counting General Dominating Set Framework},
author = {Jiayi Zheng and Boning Meng},
journal= {arXiv preprint arXiv:2603.14749},
year = {2026}
}