Abstract :
Let F be any family of graphs. Two graphs g1 = (V1,E1), G2 = (V2,E2) are said to have the same F-structure if there is a bijectionf: V1→ V2 such that a subset S induces a graph belonging to F in g1 iff its image f(S) induces a graph belonging to F in G2. We conjecture that, for any family F, a Berge graph is perfect iff it has the F-structure of some other perfect graph. An interesting special case of this conjecture is when F is the family of discs, i.e., chordless cycles of length at least five or their complements. In this paper we discuss these two conjectures and provide some partial results. In particular, we prove the first conjecture for F = {paw, copaw}.
