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 K 4-free graphs. Here we show how to decide this problem in polynomial time for (claw, K 4)-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 K 4-free planar graphs of maximum degree five.
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Le, V.B., Mosca, R., Müller, H. (2005). On Stable Cutsets in Claw-Free Graphs and Planar Graphs. In: Kratsch, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2005. Lecture Notes in Computer Science, vol 3787. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11604686_15
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DOI: https://doi.org/10.1007/11604686_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31000-6
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