A class of perfectly contractile graphs

Frédéric Maffray (1) , Nicolas Trotignon (2)

1 Leibniz - IMAG - Laboratoire Leibniz
2 CES - Centre d'économie de la Sorbonne

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 in 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. Ramirez-Alfonsin 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.

Mots clés: even pair; coloring algorithm; perfect graph

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Frédéric Maffray, Nicolas Trotignon. A class of perfectly contractile graphs. Journal of Combinatorial Theory, Series B, 2006, 96 (1), pp.1-19. ⟨10.1016/j.jctb.2005.06.011⟩. ⟨hal-00166912⟩