The Counting General Dominating Set Framework

Jiayi Zheng; Boning Meng

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arXiv · 2603.14749

The Counting General Dominating Set Framework

Jiayi Zheng·Boning Meng

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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.

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Jiayi Zheng, Boning Meng. 2026-03-16. The Counting General Dominating Set Framework. https://arxiv.org/abs/2603.14749

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