A new method for proving dominating uniqueness of graphs

Abstract

Let G be a graph of order n. A subset S of $V\left(G\right)$ is a dominating set of G if every vertex in $V\left(G\right)⧹S$ is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial $D\left(G\text{,}x\right)={\sum }_{i=\gamma \left(G\right)}^{n}d\left(G\text{,}i\right)x^i$, where $d\left(G\text{,}i\right)$ is the number of dominating sets of G of size i, and $\gamma \left(G\right)$ 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 $D\left(G\text{,}x\right)=D\left(H\text{,}x\right)$. A graph G is said to be dominating unique, or simply $D$-unique, if $D\left(H\text{,}x\right)=D\left(G\text{,}x\right)$ implies that $H\cong G$. 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 $\bar{G}\vee K_m$ is dominating unique for each $m⩾0$. As a by-product, the dominating uniqueness of many families of dense graphs is also determined.