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.
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