Friday, January 24, 2003

An Efficient Algorithm for Testing whether a Graph is Perfect

Last year we saw the resolution of the Strong Perfect Graph Conjecture, a major result in graph theory. The conjecture stated that a graph is perfect if and only if it does not contain an induced subgraph H such that H or its complement is an odd cycle of length at least five. Chudnovsky, Robertson, Seymour, and Thomas proved the conjecture (now called the Strong Perfect Graph Theorem) last spring. Vašek Chvátal has a web page describing this result.

Why am I mentioning this result here? The problem of testing whether a graph is perfect in polynomial-time remained open even after this theorem as neither characterization gives an obvious algorithm. I just saw the the abstract of a talk that Paul Seymour will give at the Institute of Advanced Study next week. There he claims he and Chudnovsky have found a polynomial-time algorithm for testing whether a graph is perfect. I cannot find more about the algorithm on the web and I will miss the talk at the institute. I will post more information about this exciting new development when I have more details.

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