Abstract :
Everett et al. (Discrete Math. 165–166 (1997) 233–252) defined a stretcher to be a graph whose edge set can be partitioned into two disjoint triangles and three vertex disjoint paths, each with an endpoint in both triangles. They also conjectured that graphs with no odd hole, no antihole and no stretcher (called Artemis graphs) may be reduced to a clique by successive contractions of even pairs. To date, no proof exists that Artemis graphs really have even pairs. We enquire here about sufficient conditions for a non-even pair of vertices to extend to a stretcher and deduce two results: the first one is a property of minimal imperfect graphs, the second one guarantees the existence of an even pair in certain Artemis graphs.
