Abstract
A safe set of a graph \(G=(V,E)\) is a non-empty subset S of V such that for every component A of G[S] and every component B of \(G[V \setminus S]\), we have \(|A| \ge |B|\) whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity of the problem. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between tree-depth and vertex cover number.
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Notes
- 1.
In the following, we (ab)use simpler notation \(\psi (A_{F},A_{F'})\) instead of \(\psi (\{A_{F},A_{F'}\})\).
- 2.
After the submission of the conference version, together with Hirotaka Ono, we found that the problem is NP-hard for these graph classes.
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Águeda, R. et al. (2016). Safe Sets in Graphs: Graph Classes and Structural Parameters. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_18
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DOI: https://doi.org/10.1007/978-3-319-48749-6_18
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