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The Counting General Dominating Set Framework

Jiayi Zheng, Boning Meng

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.

The Counting General Dominating Set Framework

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 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.
Paper Structure (27 sections, 37 theorems, 33 equations, 5 figures)

This paper contains 27 sections, 37 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. History of Domination-Type Problems
  3. Overview
  4. Organization
  5. Preliminaries
  6. Domains and Signatures
  7. The Holant Framework
  8. Gadget Construction in Holant
  9. Polynomial Interpolation
  10. The GDS Framework
  11. Definition
  12. Equivalences with Holant
  13. Writing It Apart
  14. From Gadget Construction to #Dominating Set
  15. Gadget Construction in #GDS
  16. ...and 12 more sections

Key Result

Theorem 1

#3PBS-Dominating Set,We use 3- or 3r-, P- or Pl-, B-, S- to denote the restriction of the problem to 3-regular, planar, bipartite and simple graphs respectively throughout the paper. the problem of counting dominating sets for 3-regular planar bipartite simpleWe emphasize "simple", because allowing

Figures (5)

  • Figure 1: An example gadget. Diamonds are external vertices ($V_E$), squares are bridging vertices ($V_B$), and circles are internal vertices ($V_I$).
  • Figure 2: A simple gadget.
  • Figure 3: Left: The ladder gadget. Right: The structure represented by the triangle in the ladder gadget. This is our go-to method of construction for reducing to 3-regular bipartite simple graphs, taking inspiration from xia2006vertexcover.
  • Figure 4: The reduction from counting vertex covers to counting dominating sets. Thick edges in the left graph are from a perfect matching.
  • Figure 6: The recursive gadget (left) and the starter gadget (right).

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Definition 3: $\textup{Holant}$ problems cai2017complexity
  • Lemma 4
  • Lemma 5
  • Definition 6: $\textup{\#GDS}$ problems
  • Proposition 7: $\textup{Holant}$ to $\textup{\#GDS}$
  • Proposition 8: $\textup{\#GDS}$ to domain-4 $\textup{Holant}$
  • Proposition 9
  • Proposition 10
  • ...and 31 more