Claw-free graphs are not universal fixers

For any permutation π of the vertex set of a graph G , the generalized prism π G is obtained by joining two copies of G by the matching { u π ( u ) : u ∈ V ( G ) } . Denote the domination number of G by γ ( G ) . If γ ( π G ) = γ ( G ) for all π , then G 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.