On vertex neighborhood in minimal imperfect graphs

Lubiw [11] conjectures that in a minimal imperfect Berge graph, the neighborhood graph N (v) of any vertex v must be connected; this conjecture implies a well known Chvátal’s Conjecture [6] which states that N (v) must contain a P4. In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P4-free if, and only if, every induced subgraph with at least two vertices either is disconnected or its complement is disconnected; this characterization implies that P4-free graphs can be constructed by complete join and disjoint union from isolated vertices. We propose to replace P4-free graphs by a similar construction using bipartite graphs instead of isolated vertices.