MY FAVORITE THEOREM CHARACTERIZATIONS OF PERFECT GRAPHS.pdfVIP

Konrad-Zuse-Zentrum fu? r Informationstechnik Berlin Takustra?e 7 D-14195 Berlin-Dahlem Germany MARTIN GRO¨TSCHEL My Favorite Theorem: Characterizations of Perfect Graphs Preprint SC 99-17 (June 1999) MY FAVORITE THEOREM: CHARACTERIZATIONS OF PERFECT GRAPHS MARTIN GRO¨TSCHEL The favorite topics and results of a researcher change over time, of course. One area that I have always kept an eye on is that of perfect graphs. These graphs, in- troduced in the late 50s and early 60s by Claude Berge, link various mathematical disciplines in a truly unexpected way: graph theory, combinatorial optimization, semidefinite programming, polyhedral and convexity theory, and even informati- on theory. This is not a survey of perfect graphs. It’s just an appetizer. To learn about the origins of perfect graphs, I recommend to read the historical papers [1] and [2]. The book [3] is a collection of important articles on perfect graphs. Algorithmic aspects of perfect graphs are treated in [13]. A comprehensive survey of graph classes, including perfect graphs, can be found in [5]. Hundreds of classes of perfect graphs are known, 96 important classes and the inclusion relations among them are described in [16]. So, what is a perfect graph? Complete graphs are perfect, bipartite, interval, com- parability, triangulated, parity, and unimodular graphs are perfect as well. The following beautiful perfect graph is the line graph of the complete bipartite graph K3. Due to the evolution of the theory, definitions of perfection (and versions thereof) have changed over time. To keep this article short, I do not follow the historical 1 development of the notation. I use definitions that streamline the presentation. Berge defined is aperfect graph, if and only if (1) u(G) = x G 1 ) for all node-induced subgraphs G C G, where co(G) denotes the clique number of (= largest cardinality of a clique of , i.e., a set of mutually adjacent nodes) and G) is the chromatic num