On stable cutsets in claw-free graphs and planar graphs

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.