A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one.The "strong perfect graph conjecture" (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornuéjols and Vušković -that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge's conjecture cannot have either of these properties).In this paper we prove both of these conjectures.

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