We answer a question of Brandstädt et al. by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is NP-complete. Conversely, the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently. The proof of our NP -completeness result is based on a refinement on a result due to Chvátal that recognizing decomposable graphs with maximum degree 4 is an NP-complete problem. Here, a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v. We also discuss some open problems on stable cutsets.
CITATION STYLE
Bang Le, V., & Randerath, B. (2001). On stable cutsets in line graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2204, pp. 263–271). Springer Verlag. https://doi.org/10.1007/3-540-45477-2_24
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