A proof by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas was announced in 2002[1] and published by them in 2006.
A minimally imperfect Berge graph cannot have any of these decompositions, from which it follows that no counterexample to the theorem can exist.
[6] The four types of decompositions considered in this proof are 2-joins, complements of 2-joins, balanced skew partitions, and homogeneous pairs.
When a graph has a 2-join, it may be decomposed into induced subgraphs called "blocks", by replacing one of the two subsets of vertices by a shortest path within that subset that connects one of the two complete bipartite graphs to the other; when no such path exists, the block is formed instead by replacing one of the two subsets of vertices by two vertices, one for each complete bipartite subgraph.
Therefore, if a minimally imperfect graph has a 2-join, it must equal one of its blocks, from which it follows that it must be an odd cycle and not Berge.
The full conjecture is a corollary of the strong perfect graph theorem.
The proof that every Berge graph falls into one of the five basic classes or has one of the four types of decomposition follows a case analysis, according to whether certain configurations exist within the graph: a "stretcher", a subgraph that can be decomposed into three induced paths subject to certain additional constraints, the complement of a stretcher, and a "proper wheel", a configuration related to a wheel graph, consisting of an induced cycle together with a hub vertex adjacent to at least three cycle vertices and obeying several additional constraints.