Article Dans Une Revue Discrete Applied Mathematics Année : 2023

Locating Dominating Sets in local tournaments

Résumé

A dominating set in a directed graph is a set of vertices S such that all the vertices that do not belong to S have an in-neighbour in S. A locating set S is a set of vertices such that all the vertices that do not belong to S are characterized uniquely by the in-neighbours they have in S, i.e. for every two vertices u and v that are not in S, there exists a vertex sS that dominates exactly one of them. The size of a smallest set of a directed graph D which is both locating and dominating is denoted by γLD(D). Foucaud, Heydarshahi and Parreau proved that any twin-free digraph D satisfies γLD(D)4n5+1 but conjectured that this bound can be lowered to 2n3. The conjecture is still open. They also proved that if D is a tournament, i.e. a directed graph where there is one arc between every pair of vertices, then γLD(D)n2. The main result of this paper is the generalization of this bound to connected local tournaments, i.e. connected digraphs where the in- and out-neighbourhoods of every vertex induce a tournament. We also prove γLD(D)2n3 for all quasi-twin-free digraphs D that admit a supervising vertex (a vertex from which any vertex is reachable). This class of digraphs generalizes twin-free acyclic graphs, the most general class for which this bound was known.

Dates et versions

hal-03417187 , version 1 (05-11-2021)

Identifiants

Citer

Thomas Bellitto, Caroline Brosse, Benjamin Lévêque, Aline Parreau. Locating Dominating Sets in local tournaments. Discrete Applied Mathematics, 2023, 337, pp.14-24. ⟨10.1016/j.dam.2023.04.010⟩. ⟨hal-03417187⟩
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