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. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between the tree-depth and the vertex cover number. We then conclude the paper by showing hardness for split graphs and planar graphs.
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Notes
In the following, we (ab)use simpler notation \(\psi (A_{F},A_{F'})\) instead of \(\psi (\{A_{F},A_{F'}\})\).
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Partially supported by NKFIH Grant Number 116095 (to ZsT), and MEXT/JSPS KAKENHI Grant Numbers 15K04979 (to SF), 16K05260 (to TS), 24106004 (to HO and YO), 24220003 (to HO), 26400185 (to TS), 26540005 (to HO).
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Águeda, R., Cohen, N., Fujita, S. et al. Safe sets in graphs: Graph classes and structural parameters. J Comb Optim 36, 1221–1242 (2018). https://doi.org/10.1007/s10878-017-0205-2
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DOI: https://doi.org/10.1007/s10878-017-0205-2