On stable cutsets in claw-free graphs and planar graphs

  • Authors:

  • Van Bang Le;Raffaele Mosca;Haiko Müller

  • Affiliations:

  • Institut für Informatik, Universität Rostock, 18051 Rostock, Germany;Dipartimento di Scienze, Universitá degli Studi "G.D'Annunzio", Viale Pindaro 42, Pescara 65127, Italy;School of Computing, University of Leeds, Leeds, LS2 9JT, UK

  • Venue:
  • Journal of Discrete Algorithms
  • Year:
  • 2008

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Abstract

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let K"4 and K"1","3 (claw) denote the complete (bipartite) graph on 4 and 1+3 vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a K"4-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, K"4)-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs. Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for K"4-free planar graphs with maximum degree five.