Safe sets and in-dominating sets in digraphs

Yandong Bai, Jørgen Bang-Jensen, Shinya Fujita*, Hirotaka Ono, Anders Yeo

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

A non-empty subset S of the vertices of a digraph D is a safe set if (i) for every strongly connected component M of D−S, there exists a strongly connected component N of D[S] such that there exists an arc from M to N; and (ii) for every strongly connected component M of D−S and every strongly connected component N of D[S], we have |M|≤|N| whenever there exists an arc from M to N. In the case of acyclic digraphs a set X of vertices is a safe set precisely when X is an in-dominating set, that is, every vertex not in X has at least one arc to X. We prove that, even for acyclic digraphs which are traceable (have a hamiltonian path) it is NP-hard to find a minimum cardinality safe (in-dominating) set. Then we show that the problem is also NP-hard for tournaments and give, for every positive constant c, a polynomial algorithm for finding a minimum cardinality safe set in a tournament on n vertices in which no strong component has size more than clog(n). Under the so called Exponential Time Hypothesis (ETH) this is close to best possible in the following sense: If ETH holds, then, for every ϵ>0 there is no polynomial time algorithm for finding a minimum cardinality safe set for the class of tournaments in which the largest strong component has size at most log1+ϵ(n). We also discuss bounds on the cardinality of safe sets in tournaments.

Original languageEnglish
JournalDiscrete Applied Mathematics
Volume346
Pages (from-to)215-227
Number of pages13
ISSN0166-218X
DOIs
Publication statusPublished - 31. Mar 2024

Keywords

  • In-dominating set
  • NP-complete
  • Polynomial algorithm
  • Safe set
  • Tournament

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