On $\mathcal{D}$-equivalence classes of some graphs

TL;DR

This paper investigates the domination polynomial of certain graphs, showing that barbell graphs are not unique in their domination polynomial and identifying other graphs sharing the same polynomial.

Contribution

It proves non-uniqueness of domination polynomials for barbell graphs and characterizes their $ ext{D}$-equivalence classes, including the complement of book graphs and unions of complete graphs.

Findings

Barbell graphs are not $ ext{D}$-unique for all $n \\geq 2$.

The $ ext{D}$-equivalence class of $Bar_n$ includes the complement of the book graph $B_{n-1}^c$.

Multiple families of graphs share the same domination polynomial as unions of complete graphs.

Abstract

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$ vertices, is formed by joining two copies of a complete graph $K_n$ by a single edge. We prove that for every $n\geq 2$, $Bar_n$ is not $\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. More precisely, we show that for every $n$, the $\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many graphs, which one of them is the complement of book graph of order $n-1$, $B_{n-1}^c$. Also we present many families of graphs in $\mathcal{D}$-equivalence class of $K_{n_1}\cup K_{n_2}\cup \cdots\cup K_{n_k}$.