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As an extension to the question posed recently by Bulatov, I wonder what are the maximal sub-classes of perfect graphs for which we know of combinatorial algorithms to compute a maximum independent set. Thanks in advance.

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The wikipedia article says that for all perfect graphs there is a polynomial algorithm for the graph coloring problem, maximum clique problem, and maximum independent set problem. Is this you are looking for? – Marc Bury Mar 5 '11 at 0:04
Not quite. The algorithms known to solve maximum independent set use semidefinite programming. But, for a sub-class of perfect graphs, say chordal graphs, there is a simple combinatorial algorithm based on perfect-elimination orders. It is an open question to come up with a combinatorial algorithm to compute IS of perfect graphs. So the question is what are the maximal, by inclusion for which such algorithms are known -- that are purely combinatorial. – ipsofacto Mar 5 '11 at 1:02
When you refer to other questions next time, please include links to make it easier to follow. Don’t be lazy! – Tsuyoshi Ito Mar 5 '11 at 1:24
I've recently thought about the same question, but haven't managed to google a list. Perhaps ISGCI (Information System on Graph Classes and their Inclusions) may be helpful. Here,, are all known maximal subclasses of perfect graphs, including whether the IS problem is in P or NP-C (with a reference to a paper from where one could dig out what algorithm has been used). – Standa Zivny Mar 5 '11 at 13:46
Some of the responses to… might also be useful, for instance Ernst de Ridder's explanation of some of the less well known features of the ISGCI Java tool. – András Salamon Mar 6 '11 at 21:38
up vote 4 down vote accepted

One springs to mind and is listed as a maximal subclass in ISGCI, which surprised me: perfect claw-free graphs (a.k.a. perfect quasi-line graphs). This was done by Minty for all claw-free graphs around 1980. But a couple of other algorithms for claw-free graphs, one recently in SODA 2011 that is $O(n^3)$ by Faenza, Oriolo, and Stauffer, use the Chudnovsky-Seymour structural characterization of these graphs to reduce the problem to line graphs (and therefore maximum matching) in a fairly straightforward way. If you're only looking at perfect claw-free graphs, then the earlier characterization by Maffray and Reed is sufficient (and the reduction to line graphs is more obvious).

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This sounds very interesting. thanks. – ipsofacto Mar 12 '11 at 2:22

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