Regular graphs are not universal fixers

For any permutation π of the vertex set of a graph G , the graph π G is obtained from two copies G ′ and G ″ of G by joining u ∈ V ( G ′ ) and v ∈ V ( G ″ ) if and only if v = π ( u ) . Denote the domination number of G by γ ( G ) . For all permutations π of V ( G ) , γ ( G ) ≤ γ ( π G ) ≤ 2 γ ( G ) . If γ ( π G ) = γ ( G ) for all π , then G is called a universal fixer. We prove that regular graphs and graphs with γ = 4 are not universal fixers.