Subset-Sum Representations of Domination Polynomials

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.