Subset-Sum Representations of Domination Polynomials

TL;DR

This paper explores new subset-sum representations of the domination polynomial of a graph, linking it to spanning bipartite subgraphs and conformal subgraphs, providing novel combinatorial insights.

Contribution

It introduces novel sum representations of the domination polynomial involving spanning bipartite and conformal subgraphs, advancing understanding of graph domination.

Findings

Representation of D(G,x) as sum over spanning bipartite subgraphs

Number of dominating sets expressed via conformal subgraphs

New combinatorial formulas for domination polynomials

Abstract

The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the edge and vertex set of G. One of our main results is a representation of D(G,x) as a sum ranging over spanning bipartite subgraphs of G. We call a graph G conformal if all of its components are of even order. We show that the number of dominating sets of G equals a sum ranging over vertex-induced conformal subgraphs of G.