Abstract
Independent Set Reconfiguration is one of the most well-studied problems in the setting of combinatorial reconfiguration. It is known that the problem is PSPACE-complete even for graphs of bounded bandwidth. This fact rules out the tractability of parameterizations by most well-studied structural parameters as most of them generalize bandwidth. In this paper, we study the parameterization by modular-width, which is not comparable with bandwidth. We show that the problem parameterized by modular-width is fixed-parameter tractable under all previously studied rules \({\mathsf {TAR}}\), \({\mathsf {TJ}}\), and \({\mathsf {TS}}\). The result under \({\mathsf {TAR}}\) resolves an open problem posed by Bonsma (J Graph Theory 83(2):164–195, 2016).
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Notes
Recall that an algorithm for a parameterized problem is an FPT algorithm if it runs in time \(O(f(k) \cdot n^{O(1)})\), where n is the input size, k is the parameter, and f is some computable funciton.
The \(O^*(\cdot )\) notation suppresses factors polynomial in the input size.
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Acknowledgements
Partially supported by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA) Project GRAPA 38593YJ, and by JSPS/MEXT KAKENHI Grant Numbers JP24106004, JP17H01698, JP18K11157, JP18K11168, JP18K11169, JP18H04091, JP18H06469. A preliminary version appeared in the proceedings of the 45th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2019), Lecture Notes in Computer Science 11789 (2019) 285–297.
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Belmonte, R., Hanaka, T., Lampis, M. et al. Independent Set Reconfiguration Parameterized by Modular-Width. Algorithmica 82, 2586–2605 (2020). https://doi.org/10.1007/s00453-020-00700-y
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DOI: https://doi.org/10.1007/s00453-020-00700-y