
Domination and location in twinfree digraphs
A dominating set D in a digraph is a set of vertices such that every ver...
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Counting graph orientations with no directed triangles
Alon and Yuster proved that the number of orientations of any nvertex g...
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Decomposing and colouring some locally semicomplete digraphs
A digraph is semicomplete if any two vertices are connected by at least ...
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On guarding polygons with holes
There is an old conjecture by Shermer <cit.> that in a polygon with n ve...
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Tight Bound on Vertex Cut Sparsifiers in Directed Acyclic Graphs
For an unweighted graph on k terminals, Kratsch and Wahlström constructe...
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The Local Structure of Bounded Degree Graphs
Let G=(V,E) be a simple graph with maximum degree d. For an integer k∈ℕ,...
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On the Diffusion Geometry of Graph Laplacians and Applications
We study directed, weighted graphs G=(V,E) and consider the (not necessa...
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Locating Dominating Sets in local tournaments
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 inneighbour 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 inneighbours 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 twinfree 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 outneighbourhoods of every vertex induce a tournament. We also prove γ^LD(D)≤2n/3 for all quasitwinfree digraphs D that admit a supervising vertex (a vertex from which any vertex is reachable). This class of digraphs generalizes twinfree acyclic graphs, the most general class for which this bound was known.
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