Domination and location in twin-free digraphs

Foucaud, Florent; Heydarshahi, Shahrzad; Parreau, Aline

doi:10.1016/j.dam.2020.03.025

Cited by 11 publications

(18 citation statements)

References 13 publications

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“…Let D be a digraph and u be a non-isolated vertex of D. Then V (D)\ {u} is a locating-dominating set of D. In particular, for any directed graph containing at least one edge, Γ d (D) ≤ n − 1. In [13], the authors have characterized digraphs that reach this extremal value. This characterization will be useful for studying the extremal values of → γ LD (G) and → Γ LD (G).…”

Section: Preliminary Results and Examplesmentioning

confidence: 99%

“…Thus S must induce a locating-dominating set in K m . Since K m is oriented in a transitive way, by [13], we necessarily have at least ⌈m/2⌉ vertices in V (K m ) ∩ S and so in S.…”

Section: Basic Resultsmentioning

confidence: 99%

“…Location-domination in directed graphs was briefly mentioned in several articles (see e.g. [7,28]) and further studied in [13]. A prove that if G is a C 4 -free bipartite graph (which in particular, contains the class of trees), then → Γ LD (G) = α(G) where α(G) is the maximum size of an independent set.…”

Section: Introductionmentioning

confidence: 99%

“…Somehow surprisingly at first glance, → Γ LD (G) ≥ γ LD (G) is not always true. In [13], Foucaud et al have shown that for a complete graph K n on n vertices we have…”

Section: Introductionmentioning

confidence: 99%

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Locating-dominating sets: from graphs to oriented graphs

Bousquet¹,

Deschamps²,

Lehtilä³

et al. 2021

Preprint

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A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u, v / ∈ S, the neighbourhood of u and v on S are distinct (i.e. N (u) ∩ S = N (v) ∩ S). Locating-dominating sets have received a considerable attention in the last decades. In this paper, we consider the oriented version of the problem. A locating-dominating set of an oriented graph is a set S such that for every u, v ∈ V \ S, N − (u) ∩ S = N − (v) ∩ S. We consider the following two parameters. Given an undirected graph G, we look forwhich is the size of the smallest (largest) optimal locating-dominating set over all orientations of G. In particular, if D is an orientation ofwhere γLD(D) is the minimum size of a locatingdominating code of D.For the best orientation, we prove that, for every twin-free graph G on n vertices, → γ LD (G) ≤ n/2 which proves a "directed version" of a widely studied conjecture on location-domination number. Moreover, we give some bounds for → γ LD (G) on many graph classes and drastically improve value n/2 for (almost) d-regular graphs by showing that → γ LD (G) ∈ O(log d/d • n) using a probabilistic argument. While → γ LD (G) ≤ γLD(G) holds for every graph G, we give some graph classes such as outerplanar graphs for which → ΓLD(G) ≥ γLD(G) and some for which → ΓLD(G) ≤ γLD(G) such as complete graphs. We also give general bounds for → ΓLD(G) such as → ΓLD(G) ≥ α(G). Finally, we show that for many graph classes → ΓLD(G) is polynomial on n but we leave open the question whether there exist graphs with → ΓLD(G) ∈ O(log n). * This work was supported by ANR project GrR (ANR-18-CE40-0032) † Research supported by the Finnish cultural foundation locating-dominating set of a directed graph D is a subset S of its vertices such that two vertices not in S have distinct and non-empty in-neighbourhoods in S. The directed location-domination number of D, denoted by γ LD (D), is the size of a smallest locating-dominating set of D.Two oriented graphs with the same underlying graph can have a very different behaviour towards locating-dominating sets. Let us illustrate it on tournaments that are oriented complete graphs. Transitive tournaments (i.e. acyclic tournaments) have directed location-domination number ⌈n/2⌉ whereas one can construct locating-dominating sets of size ⌈log n⌉ for a well-chosen orientation of K n [28]. Following the idea of Caro and Henning for domination [6] and the work started by Skaggs [28], we study in this paper the best and worst orientations of a graph for locating-dominating sets. A similar line of work has been recently initiated for the related concepts of identifying codes [9] and metric dimension [2].The two parameters that will be considered in this paper are the following. The lower directed location-domination number of an undirected graph G, denoted by → γ LD (G), is the minimum directed location-domination number over all the orientations of G. The upper directed locationdomination number of an undirected graph G, denoted by → Γ LD (G), is the maximum ...

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Section: Preliminary Results and Examplesmentioning

confidence: 99%

“…Thus S must induce a locating-dominating set in K m . Since K m is oriented in a transitive way, by [13], we necessarily have at least ⌈m/2⌉ vertices in V (K m ) ∩ S and so in S.…”

Section: Basic Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Somehow surprisingly at first glance, → Γ LD (G) ≥ γ LD (G) is not always true. In [13], Foucaud et al have shown that for a complete graph K n on n vertices we have…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Locating-dominating sets: from graphs to oriented graphs

Bousquet¹,

Deschamps²,

Lehtilä³

et al. 2021

Preprint

Self Cite

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show abstract

“…Thus, [20] can be seen as an attempt to extend undirected distance-hereditary graphs to directed distance-hereditary graphs. In [9], twins have been defined to obtain results about domination and location-domination, and in [17] (see also [5, p. 282]) in studying diameter in digraphs. In [22] twins are introduced in context of a distance based directed version of distance-hereditary graphs, but they do not lead to a characterization of this graph class.…”

Section: Directed Distance-hereditary Graphsmentioning

confidence: 99%

Twin-Distance-Hereditary Digraphs

Komander¹,

Rehs²

2021

Preprint

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We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define directed twin-distance-hereditary graphs, which can be constructed by several twin and pendant vertex operations analogously to undirected distance-hereditary graphs and which still preserves the distance hereditary property. We give a characterization by forbidden induced subdigraphs and place the class in the hierarchy, comparing it to related classes. We further show algorithmic advantages concerning directed width parameters, directed graph coloring and some other well-known digraph problems which are NP-hard in general, but computable in polynomial or even linear time on twin-distance-hereditary digraphs. This includes computability of directed path-width and tree-width in linear time and the directed chromatic number in polynomial time. From our result that directed twin-distance-hereditary graphs have directed clique-width at most 3 it follows by Courcelle's theorem on directed clique-width that we can compute every graph problem describable in which is describable in monadic second-order logic on quantification over vertices and vertex sets as well as some further problems like Hamiltonian Path/Cycle in polynomial time.

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On Finding the Best and Worst Orientations for the Metric Dimension

et al. 2023

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The (directed) metric dimension of a digraph D, denoted by − − → MD(D), is the size of a smallest subset S of vertices such that every two vertices of D are distinguished via their distances from the vertices in S.In this paper, we investigate the graph parameters BOMD(G) and WOMD(G) which are respectively the smallest and largest metric dimension over all orientations of G. We show that those parameters are related to several classical notions of graph theory and investigate the complexity of determining those parameters. We show that BOMD(G) = 1 if and only if G is hypotraceable (that is has a path spanning all vertices but one), and deduce that deciding whether BOMD(G) ≤ k is NP-complete for every positive integer k. We also show that WOMD(G) ≥ α(G) − 1, where α(G) is the stability number of G. We then deduce that for every fixed positive integer k, we can decide in polynomial time whether WOMD(G) ≤ k.The most significant results deal with oriented forests. We provide a linear-time algorithm to compute the metric dimension of an oriented forest and a linear-time algorithm that, given a forest F , computes an orientation D − with smallest metric dimension (i.e. such that − − → MD(D − ) = BOMD(F )) and an orientation D + with largest metric dimension (i.e. such that − − → MD(D + ) = WOMD(F )).

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Domination and location in twin-free digraphs

Cited by 11 publications

References 13 publications

Locating-dominating sets: from graphs to oriented graphs

Locating-dominating sets: from graphs to oriented graphs

Twin-Distance-Hereditary Digraphs

On Finding the Best and Worst Orientations for the Metric Dimension

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