Graphs with C3-free vertices are not universal fixers

Abstract

A non-isolated vertex x ϵ V(G) is called C3-free if x belongs to no triangle of G. In [1] 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 φ : V(G)→ V(G¹), we define the prism φG of G as follows: V(φG) = V(G) U V(G') and E(φG) = E(G) U E(G) U Mφ, where Mφ = {uφ(u) : u ϵ V(G)}. Let γ(G) be the domination number of G. If γ(φG) = γ(G) for any bijective function φ, then G is called a universal fixer. In [3] it is conjectured that the only universal fixer is the edgeless graph K_n In this note, we prove that any graph G with C₃-free vertices is not a universal fixer graph. © 2017 Utilitas Mathematica Publishing Inc.. All rights reserved.