Graphs with $C_3_-free vertices are not universal fixers

Abstract

A non-isolated vertex $x\in V(G)$ is called $C_{3}$-free if $x$ belongs to no triangle of $G$. In \cite{BMW} Burger, Mynhardt and Weakley introduced the idea of universal fixers. Let $G=(V,E)$ be a graph with $n$ vertices and $G'$ a copy of $G$. For a bijective function $\pi:V(G)\mapsto V (G')$, we define the prism $\pi G$ of $G$ as follows: $V(\pi G)=V(G)\cup V(G')$ and $E(\pi G)=E(G)\cup E(G')\cup M_{\pi}$, where $M_{\pi}=\{u\pi (u): u\in V(G)\}$. Let $\gamma(G)$ be the domination number of $G$. If $\gamma(\pi G)=\gamma(G)$ for any bijective function $\pi$, then $G$ is called a universal fixer. In \cite{MX} it is conjectured that the only universal fixer is the edgeless graph $\bar{K_n}$. In this note, we prove that any graph $G$ with $C_3$-free vertices is not a universal fixer graph.