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 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.

翻译：我们引入了一个新的计数问题框架，称为#GDS，它涵盖了#$(σ, ρ)$-Set——一类支配型问题，包括计数支配集和计数全支配集。我们探讨了#GDS与著名的Holant之间的复杂关系。我们将Holant的构件构造技术适配到#GDS框架中；利用该技术，我们证明了3-正则平面二分简单图上计数支配集的#P完全性。通过对Holant二分法的一般化，以及通过对称二分图的一种特殊归约方法，我们还证明了同一图类上计数全支配集的#P完全性。