On stable cutsets in claw-free graphs and planar graphs

Authors:
Van Bang Le;Raffaele Mosca;Haiko Müller
Affiliations:
Institut für Informatik, Universität Rostock, Rostock, Germany;Dipartimento di Scienze, Universitá degli Studi “G.D'Annunzio”, Pescara, Italy;School of Computing, University of Leeds, Leeds, UK
Venue:
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Year:
2005

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Abstract

To decide whether a line graph (hence a claw-free graph) of maximum degree five admits a stable cutset has been proven to be an nP-complete problem. The same result has been known for K4-free graphs. Here we show how to decide this problem in polynomial time for (claw, K4)-free graphs and for a claw-free graph of maximum degree at most four. As a by-product we prove that the stable cutset problem is polynomially solvable for claw-free planar graphs, and for planar line graphs. now, the computational complexity of the stable cutset problem restricted to claw-free graphs and claw-free planar graphs is known for all bounds on the maximum degree. Moreover, we prove that the stable cutset problem remains nP-complete for K4-free planar graphs of maximum degree five.