Some classes of strongly perfect graphs

Abstract

A graph is perfect if for each of its induced subgraphs H, the chromatic number of H is equal to the maximum number of mutually adjacent vertices in it. A graph is ‘strongly perfect’ if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it. Every strongly perfect graph is perfect but the converse is generally not valid. For example, the complement of an even cycle of length at least 6 is not strongly perfect though it is perfect. The strongly perfect graphs form an interesting class of perfect graphs, because the complement of a strongly perfect graph is not necessarily strongly perfect, unlike the case with the perfect graphs and their relevance to the famous Berge's strong perfect graph conjecture which has been eluding a solution for more than three decades. Meyniel graphs, line graphs that are free from some graphs, comparability graphs, costrongly perfect graphs are some of the most important classes of strongly perfect graphs. Here, we summarize the results concerning strongly perfect graphs.