# Safe sets in digraphs

A non-empty subset S of the vertices of a digraph D is called 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 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 log^1+ϵ(n). We also discuss bounds on the cardinality of safe sets in tournaments.

## Authors

• 1 publication
• 6 publications
• 1 publication
• 10 publications
04/14/2019

### Linear algorithms on Steiner domination of trees

A set of vertices W in a connected graph G is called a Steiner dominatin...
04/01/2019

### Complexity and Algorithms for Semipaired Domination in Graphs

For a graph G=(V,E) with no isolated vertices, a set D⊆ V is called a se...
09/05/2021

### A linear-time algorithm for semitotal domination in strongly chordal graphs

In a graph G=(V,E) with no isolated vertex, a dominating set D ⊆ V, is c...
08/31/2020

### Semipaired Domination in Some Subclasses of Chordal Graphs

A dominating set D of a graph G without isolated vertices is called semi...
02/04/2019

### Safe projections of binary data sets

Selectivity estimation of a boolean query based on frequent itemsets can...
04/21/2021

### QCSP on Reflexive Tournaments

We give a complexity dichotomy for the Quantified Constraint Satisfactio...
05/22/2017

### Classification Using Proximity Catch Digraphs (Technical Report)

We employ random geometric digraphs to construct semi-parametric classif...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.