[Paper Review] Counting Dominating Sets of Graphs

Irene Heinrich, Peter Tittmann|arXiv (Cornell University)|Jan 12, 2017

Topological and Geometric Data Analysis5 references3 citations

TL;DR

This paper establishes a precise formula for counting dominating sets in any finite simple graph using the number of complete bipartite subgraphs in its complement. It proves that the total number of dominating sets, d(G), equals 2ⁿ − 1 + 2[a(G) − b(G)], where a(G) and b(G) count even-even and odd-odd complete bipartite subgraphs K₂ₚ,₂ₚ and K₂ₚ₊₁,₂ₚ₊₁ in the complement graph. This result provides a new algebraic characterization of domination and offers a novel proof that the number of dominating sets in any finite graph is always odd.

ABSTRACT

Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborhood polynomial of a graph.

Motivation & Objective

• To establish a direct algebraic relationship between the number of dominating sets in a graph and the structure of complete bipartite subgraphs in its complement.
• To provide a new combinatorial proof that the number of dominating sets in any finite graph is odd.
• To unify the domination polynomial and neighborhood polynomial through structural invariants of the complement graph.
• To derive a closed-form expression for the domination polynomial using inclusion-exclusion and Möbius inversion on edge subsets.

Proposed method

• The authors use the neighborhood complex of a graph and its generating function, the neighborhood polynomial, to analyze subsets contained in open neighborhoods of vertices.
• They apply inclusion-exclusion principles to express the neighborhood polynomial as a signed sum over intersections of neighborhoods of vertex subsets.
• They derive a generating function identity for the neighborhood polynomial via Möbius inversion over edge subsets of the graph.
• They establish a duality between dominating sets in G and neighborhoods in the complement graph G̅, showing D(G,1) + N(G̅,1) = 2ⁿ.
• They use the structure of complete bipartite subgraphs Kₚ,ₚ in G̅ to compute N(G̅,1), distinguishing cases by parity of p and q.
• They apply a signed sum over isomorphism types of complete bipartite subgraphs to derive the final formula: d(G) = 2ⁿ − 1 + 2[a(G) − b(G)].

Experimental results

Research questions

• RQ1 How can the number of dominating sets in a graph be expressed in terms of subgraph counts in its complement?
• RQ2 What structural properties of the complement graph determine the parity and exact count of dominating sets?
• RQ3 Can the known fact that the number of dominating sets is always odd be derived from a combinatorial formula involving subgraph enumeration?
• RQ4 What is the precise relationship between the domination polynomial and the neighborhood polynomial of a graph?
• RQ5 How do edge subsets and their inclusion-exclusion properties contribute to the computation of neighborhood and domination polynomials?

Key findings

• The number of dominating sets d(G) in a graph G of order n is given by d(G) = 2ⁿ − 1 + 2[a(G) − b(G)], where a(G) and b(G) count subgraphs isomorphic to K₂ₚ,₂ₚ and K₂ₚ₊₁,₂ₚ₊₁ in the complement graph.
• The formula confirms that d(G) is always odd, as the term 2[a(G) − b(G)] is even and 2ⁿ − 1 is odd.
• The complement graph's complete bipartite subgraphs of even and odd order contribute positively or negatively to the count, depending on the parity of their parts.
• The neighborhood polynomial of the complement graph G̅ fully determines the domination polynomial of G via the identity D(G,x) + N(G̅,x) = (1+x)ⁿ.
• The sum over edge subsets F of G of (−1)ˡᶠˡ N(G−F,x) equals zero unless G is isomorphic to a complete bipartite graph or empty, in which case it yields a closed-form expression.
• The result provides a new algebraic proof of the oddness of the number of dominating sets, resolving a known conjecture via subgraph enumeration.