Paper detail

Graphes parfaits : structure et algorithmes

This work is the PhD thesis of Nicolas Trotignon, written in 2004 under the supervision of Frédéric Maffray. It is motivated by the desire for a better understanding of perfect graphs. The proof of the Claude Berge's perfect graph conjecture in 2002 by Chudnovsky, Robertson, Seymour and Thomas has shed a new light on this field of combinatorics. But some questions are still unsettled, particulary the existence of a combinatorial algorithm for the coloring of perfect graphs. An even pair of a graph is a pair of vertices such that every path joining them has even length. As proved by Fonlupt and Uhry, the contraction of an even pair preserves the chromatic number, and when applied recursively may lead to an optimal coloring. We prove a conjecture of Everett and Reed saying that this method works for a class of perfect graphs: Artemis graphs. This yields a coloring algorithm for Artemis graphs with complexity $O(n^2m)$. We give an $O(n^9)$ algorithm for the recognition of Artemis graphs. Other recognition algorithms are also given, each of them based on subgraph detection routines for Berge graphs. We show that these subgraph detection problems are NP-complete when extended to general graphs.