Locating Dominating Sets in local tournaments
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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 s∈ S 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 γ^LD(D). Foucaud, Heydarshahi and Parreau proved that any twin-free digraph D satisfies γ^LD(D)≤4n/5 +1 but conjectured that this bound can be lowered to 2n/3. 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 γ^LD(D)≤⌈n/2⌉. 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 γ^LD(D)≤2n/3 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.
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