Regular graphs are not universal fixers

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