Locating and total dominating sets in trees

Haynes, Teresa W.; Henning, Michael A.; Howard, J

doi:10.1016/j.dam.2006.01.002

Cited by 113 publications

(87 citation statements)

References 2 publications

(2 reference statements)

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“…Locating-total domination and differentiating-total domination were introduced by Haynes et al [5]. They established bounds on these parameters in a tree and investigated the ratio of the two parameters in trees.…”

Section: Introductionmentioning

confidence: 99%

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Bounds on the locating-total domination number of a tree

Chen

Sohn

2011

Discrete Applied Mathematics

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a b s t r a c tIn this paper, we continue the study of locating-total domination in graphs, introduced by Haynes et al. [T.W. Haynes, M.A. Henning, J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics 154 (8) (2006) 1293-1300]. A total dominating set S in a graph G = (V , E) is a locating-total dominating set of G if, for every pair of distinct vertices u andThe minimum cardinality of a locating-total dominating set is the locating-total domination number γ L t (G). We show that, for a tree T of order n ≥ 3 with l leaves and s support vertices, n+l+1Moreover, we constructively characterize the extremal trees achieving these bounds.

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Section: Introductionmentioning

confidence: 99%

“…The concept of a locating-total domination in graph was introduced in [2,5]. The location of monitoring devices, such as surveillance cameras or fire alarms, to safeguard a system serves as the motivation for this work.…”

Section: Introductionmentioning

confidence: 99%

Bounds on the locating-total domination number of a tree

Chen

Sohn

2011

Discrete Applied Mathematics

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“…Applications of the metric dimension to the navigation of robots in networks are discussed in [4] and applications to chemistry in [5,6]. This invariant was studied further in a number of other papers including, for instance [7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

The Simultaneous Local Metric Dimension of Graph Families

et al. 2017

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Abstract:In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if v, y). A set S ⊆ V is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set S ⊆ V is said to be a simultaneous local metric generator for a graph family G = {G 1 , G 2 , . . . , G k }, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of G. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter.

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“…γ SD (G)), is called the differentiating-domination (resp. strict differentiatingdomination) number of G. Some of these concepts may be found in [4] and are investigated in [1], [3], [5], and [7].…”

Section: It Is a Strictly Differentiating Set If It Is Differentiatinmentioning

confidence: 99%