Claw-free graphs are not universal fixers

For any permutation ππ of the vertex set of a graph GG, the generalized prism πGπG is obtained by joining two copies of GG by the matching {uπ(u):u∈V(G)}{uπ(u):u∈V(G)}. Denote the domination number of GG by γ(G)γ(G). If γ(πG)=γ(G)γ(πG)=γ(G) for all ππ, then GG is called a universal fixer. The edgeless graphs are the only known universal fixers, and are conjectured to be the only universal fixers. We prove that claw-free graphs are not universal fixers.