Raffaele Mosca - Some results on stable sets for k-colorable P₆-free graphs and generalizations

dmtcs:579 - Discrete Mathematics & Theoretical Computer Science, August 28, 2012, Vol. 14 no. 2 - https://doi.org/10.46298/dmtcs.579
Some results on stable sets for k-colorable P₆-free graphs and generalizationsArticle

Authors: Raffaele Mosca 1

  • 1 Dipartimento di Economia [G. D'Annunzio]

This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).

Volume: Vol. 14 no. 2
Section: Graph and Algorithms
Published on: August 28, 2012
Accepted on: June 9, 2015
Submitted on: September 23, 2010
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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