On vertex neighborhood in minimal imperfect graphs

Abstract

Lubiw (J. Combin. Theory Ser. B 51 (1991) 24) 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 (Chvátal, First Workshop on Perfect Graphs, Princeton, 1993) which states that N(v) must contain a P 4 . In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P 4 -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 P 4 -free graphs can be constructed by complete join and disjoint union from isolated vertices. We propose to replace P 4 -free graphs by a similar construction using bipartite graphs instead of isolated vertices.