Two vertices x, y are siblings with respect to a set Q if both Q∪{x} and Q∪{y} induce a P4. Two graphs G = (V, E) and G′ = (V′, E′) are said to have the same sibling-structure if there is a bijection f: V → V′ such that vertices x, y are siblings with respect to a set Q in G if and only if f(x), f(y) are siblings with respect to f(Q) in G′. We prove that if two graphs have the same sibling-structure then either both graphs are perfect or both graphs are imperfect.
