Safe sets in graphs: Graph classes and structural parameters

Raquel Águeda, Nathann Cohen, Shinya Fujita, Sylvain Legay, Yannis Manoussakis, Yasuko Matsui, Leandro Montero, Reza Naserasr, Hirotaka Ono, Yota Otachi, Tadashi Sakuma, Z. Tuza, Renyu Xu

Research output: Contribution to journalArticle

5 Citations (Scopus)


A safe set of a graph (Formula presented.) is a non-empty subset S of V such that for every component A of G[S] and every component B of (Formula presented.), we have (Formula presented.) 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.

Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalJournal of Combinatorial Optimization
Publication statusAccepted/In press - Nov 17 2017


  • Graph algorithm
  • Graph class
  • Parameterized complexity
  • Safe set

ASJC Scopus subject areas

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics

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  • Cite this

    Águeda, R., Cohen, N., Fujita, S., Legay, S., Manoussakis, Y., Matsui, Y., Montero, L., Naserasr, R., Ono, H., Otachi, Y., Sakuma, T., Tuza, Z., & Xu, R. (Accepted/In press). Safe sets in graphs: Graph classes and structural parameters. Journal of Combinatorial Optimization, 1-22.