Location-domination in line graphs

Foucaud, Florent; Henning, Michael A.

doi:10.1016/j.disc.2016.05.020

Cited by 9 publications

(5 citation statements)

References 18 publications

(43 reference statements)

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“…The minimum cardinality of such a set, denoted by γ L (G), is the location-domination number of G. There is also an extensive literature on γ L (G) studying multiple aspects: complexity [13,17], specific families [14,16,25,29,31], bounds [2,15,22,40], and approximation algorithms [45]. Clearly, an LD-set is an MLD-set, and so it is also a resolving set; consequently,…”

Section: Introductionmentioning

confidence: 99%

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

González

Hernando

Mora

2018

Applied Mathematics and Computation

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A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so far.

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Section: Introductionmentioning

confidence: 99%

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

González

Hernando

Mora

2018

Applied Mathematics and Computation

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“…Charbit et al (2021) studied the edge clique cover number of graphs with independence number two, which are necessarily claw-free. They gave the first known proof of a linear bound in n for ecc(𝐺) for such graphs, improving upon the bound of 𝑂(𝑛4∕3log1∕3𝑛) .More precisely they proved that ecc(𝐺) is at most the minimum of 𝑛+𝛿(𝐺) and 2𝑛−Ω(√𝑛log𝑛),where 𝛿(𝐺) is the minimum degree of G. Foucaud et al (2017) proved two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.…”

Section: Graph Coloring Problemmentioning

confidence: 99%

Suitable Site Selection of Water ATMs (Basis of Interior/Exterior Conditions) Using Graph Theory

Ahmed

Islam

Bora

2022

Journal of Cases on Information Technology

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Maintenance performs a vital role in assuring safety operation, enhancing the quality and accumulating the durability of the system. In this paper a method has been evolved to solve the different kinds of issues like new software installation, upgradation of current software, fixing of required equipment, raw water supply problem etc. To continue the service of water ATMs we cannot start maintenance of all the water ATMs together in any particular site, so authors require a proper network planning. Authors have a developed an algorithm to select the best site for fixing of Water ATMs. So individual can use this algorithm and find out the best sites for setting up of Water ATMs such that maximum numbers of persons gain the advantage of Water ATMs. In addition authors have planned an IoT enable Water ATMs. The IoT enabled technology put in the various function of Water ATMs, safeguarding the quality of water.

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“…Such a theorem does not hold for the location-domination number of undirected graphs, for example complete graphs and stars of order n have location-domination number n − 1, see [19]. Nevertheless, Garijo, González and Márquez have conjectured in [9] that in the absence of twins, the upper bound of Ore's theorem also holds for the location-domination number of undirected graphs; they also proved that an upper bound of roughly two thirds the order holds in this context (see [4,5,6] for further developments on this matter).…”

Section: Introductionmentioning

confidence: 99%

Domination and location in twin-free digraphs

Foucaud

Heydarshahi

Parreau

2020

Discrete Applied Mathematics

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A dominating set D in a digraph is a set of vertices such that every vertex is either in D or has an in-neighbour in D. A dominating set D of a digraph is locating-dominating if every vertex not in D has a unique set of in-neighbours within D. The location-domination number γL(G) of a digraph G is the smallest size of a locating-dominating set of G. We investigate upper bounds on γL(G) in terms of the order of G. We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with the same (open or closed) in-neighbourhoods. Thus, we investigate the value of γL(G) in the absence of twins and give a general method for constructing small locating-dominating sets by the means of special dominating sets. In this way, we show that for every twin-free digraph G of order n, γL(G) ≤ 4n 5 holds, and there exist twin-free digraphs G with γL(G) = 2(n−2) 3 . If moreover G is a tournament or is acyclic, the bound is improved to γL(G) ≤ ⌈ n 2 ⌉, which is tight in both cases. 1 A set of vertices of an undirected graph G is a dominating set if it is a dominating set of the digraph obtained from G by replacing each edge by two symmetric arcs.

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Location-domination in line graphs

Cited by 9 publications

References 18 publications

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

Suitable Site Selection of Water ATMs (Basis of Interior/Exterior Conditions) Using Graph Theory

Domination and location in twin-free digraphs

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