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

Abstract.

 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. Chvátal 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 Chvátal's conjecture to the case of weakly chordal graphs. Here we focus on weakly chordal graphs, and we reduce Chvátal's conjecture to a restricted case. Our method lays out the structure of all bull-free weakly chordal graphs. These results have been used recently by Hayward to establish Chvátal's conjecture for this restricted case and therefore in full.