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Safe sets in graphs: Graph classes and structural parameters

Author

Listed:
  • Raquel Águeda

    (Universidad de Castilla-La Mancha)

  • Nathann Cohen

    (University Paris-Sud)

  • Shinya Fujita

    (Yokohama City University)

  • Sylvain Legay

    (University Paris-Sud)

  • Yannis Manoussakis

    (University Paris-Sud)

  • Yasuko Matsui

    (Tokai University)

  • Leandro Montero

    (University Paris-Sud)

  • Reza Naserasr

    (University Paris-Diderot)

  • Hirotaka Ono

    (Nagoya University)

  • Yota Otachi

    (Kumamoto University)

  • Tadashi Sakuma

    (Yamagata University)

  • Zsolt Tuza

    (MTA Rényi Institute
    University of Pannonia)

  • Renyu Xu

    (Shandong University)

Abstract

A safe set of a graph $$G=(V,E)$$ 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 $$G[V {\setminus } S]$$ G [ V \ S ] , we have $$|A| \ge |B|$$ | A | ≥ | 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.

Suggested Citation

  • Raquel Águeda & Nathann Cohen & Shinya Fujita & Sylvain Legay & Yannis Manoussakis & Yasuko Matsui & Leandro Montero & Reza Naserasr & Hirotaka Ono & Yota Otachi & Tadashi Sakuma & Zsolt Tuza & Renyu , 2018. "Safe sets in graphs: Graph classes and structural parameters," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1221-1242, November.
    • Handle: RePEc:spr:jcomop:v:36:y:2018:i:4:d:10.1007_s10878-017-0205-2
    DOI: 10.1007/s10878-017-0205-2
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