A new method for proving dominating uniqueness of graphs

Let G be a graph of order n. A subset S of V(G) is a dominating set of G if every vertex in V(G)⧹S is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial D(G,x)=∑i=γ(G)nd(G,i)xi, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the size of a smallest dominating set of G, called the domination number of G. We say two graphs G and H are dominating equivalent if D(G,x)=D(H,x). A graph G is said to be dominating unique, or simply D-unique, if D(H,x)=D(G,x) implies that H≅G. The goal of this paper is to find a new approach to determine the dominating uniqueness of graphs. In this paper, we define a new graph polynomial, called star polynomial, and introduced an analogy notion of star uniqueness of graphs. As an application, if G is a graph without isolated vertices, we show that a graph G is star unique if and only if G‾∨Km is dominating unique for each m⩾0. As a by-product, the dominating uniqueness of many families of dense graphs is also determined.