  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.
