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Perfect graph

A perfect graph is a graph such that for every induced subgraph of , the clique number equals the chromatic number, i.e., . A graph that is not a perfect graph is called an imperfect graph (Godsil and Royle 2001, p. 142).A graph for which (without any requirement that this condition also hold on induced subgraphs) is called a weakly perfect graph. All perfect graphs are therefore weakly perfect by definition.A graph is strongly perfect if every induced subgraph has an independent set meeting all maximal cliques of . While all strongly perfect graphs are perfect, the converse is not necessarily true. Since every -free graph (where is a path graph) is strongly perfect (Ravindra 1999) and every strongly perfect graph is perfect, if a graph is -free, it is perfect.Perfect graphs were introduced by Berge (1973) motivated in part by determining the Shannon capacity of graphs (Bohman 2003). Note that rather confusingly, perfect graphs are distinct..

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