Graphs with $C_3_-free vertices are not universal fixers

TL;DR

This paper proves that graphs containing vertices not part of any triangle cannot be universal fixers, supporting the conjecture that only edgeless graphs have this property.

Contribution

It establishes that graphs with $C_3$-free vertices are not universal fixers, advancing understanding of the structure of universal fixers.

Findings

Graphs with $C_3$-free vertices are not universal fixers.

Supports the conjecture that only edgeless graphs are universal fixers.

Provides a new criterion to identify non-universal fixer graphs.

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.