Graphs with C₃-free vertices are not universal fixers

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A non-isolated vertex x∈ V(G) is called C₃-free if x belongs to no triangle of G. In 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 π:V(G) V (G'), we define the prism πG of G as follows: V(πG)=V(G)∪ V(G') and E(πG)=E(G)∪ E(G')∪ Mπ, where Mπ={uπ(u): u∈ V(G)}. Let γ(G) be the domination number of G. If γ(πG)=γ(G) for any bijective function π, then G is called a universal fixer. In MX it is conjectured that the only universal fixer is the edgeless graph Kₙ. In this note, we prove that any graph G with C₃-free vertices is not a universal fixer graph.