On the Structure of Bull-Free Perfect Graphs, 2: the Weakly Chordal Case

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. A graph is perfectly orderable if it admits an ordering such that the greedy sequential method applied on this ordering produces an optimal coloring for every induced subgraph. Chv atal conjectured that every bull-free graph with no odd hole or antihole is perfectly orderable. In a previous paper we studied the structure of general bull-free perfect graphs, and reduced Chv atal's conjecture to the case of weakly triangulated graphs. Here we focus on weakly triangulated graphs. Our method lays out the structure of all bull-free weakly triangulated graphs. Its results have been used recently by Hayward to establish Chv atal's conjecture in full. 1Universidade Federal do Rio de Janeiro, Instituto de Matem atica, Caixa Postal 68530, 21944 Rio de Janeiro, RJ, Brasil. [email protected]. Partially supported by CNPq, CAPES and COFECUB. 2CNRS, Laboratoire Leibniz, 46 avenue F elix Viallet, 38031 Grenoble Cedex, France. [email protected]. Partially supported by CNPq, CAPES and COFECUB (Brazil) and by the Air Force O ce for Scienti c Research under grant number F49620-93-1-0041 to Rutgers University Center for Operations Research (New Jersey). 3PUC-Rio, Dept. de Engenharia El etrica, Rua Marquês de S~ ao Vicente 225, Predio Cardeal Leme, Sala 401, CEP 22453, Rio de Janeiro, RJ, Brasil. [email protected] 1