Relatively prime dominating polynomial in graphs

Abstract

We introduce the concept of relatively prime domination polynomial of a graph $G$. The relatively prime domination polynomial of a graph $G$ of order $n$ is the polynomial $D_{r p d}(G, x)=\sum_{k=\gamma_{p d}(G)}^n d_{r p d}(G, k) x^k$ where $d_{r p d}(G, k)$ is the number of relatively prime dominating sets of $G$ of size $k$, and $\gamma_{r p d}(G)$ is the relatively prime domination number of $G$. We compute this polynomial for path $P_n$, complete bipartite graph $K_{m, n}$, star $K_{1, n}$, bistar $B_{m, n}$, spider graph $K_{1, n, n}$ and Helm graph $H_n$.