A class of perfectly contractile graphs
Frédéric Maffray, Nicolas Trotignon · 1997
Abstract. We consider the class A of graphs that contain no odd hole, no antihole, and no “prism ” (a graph consisting of two disjoint triangles with three disjoint paths between them). We prove that every graph G ∈ A different from a clique has an “even pair ” (two vertices that are not joined by a chordless path of odd length), as conjectured by Everett and Reed [see the chapter “Even pairs ” in the book Perfect Graphs, J.L. Ramírez-Alfonsín and B.A. Reed, eds., Wiley Interscience, 2001]. Our proof is a polynomial-time algorithm that produces an even pair with the additional property that the contraction of this pair yields a graph in A. This entails a polynomial-time algorithm, based on successively contracting even pairs, to color optimally every graph in A. This generalizes several results concerning some classical families of perfect graphs. 1