Computing the k -metric dimension of graphs

Yero, Ismael G.; Estrada‐Moreno, Alejandro; Rodríguez-Velázquez, Juan A.

doi:10.1016/j.amc.2016.12.005

Cited by 28 publications

(28 citation statements)

References 20 publications

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“…For a tree T different from a path, the value of parameter ς(T ) can be computed in linear time with respect to the order of T , as it was shown in [45].…”

Section: On K-partition Dimensional Graphsmentioning

confidence: 99%

“…It can be proved that the time complexity of computing d * (G) is O(n 3 ) applying a procedure similar to that used in [45] to prove the time complexity of computing d(G). Examples of graphs that satisfy Corollary 21 are K n , C 2k+1 , where k is a positive integer, and the wheel graph W 1,5 .…”

Section: Theorem 16 [11]mentioning

confidence: 99%

“…It was shown in [45] that dim k (T ) can be computed in linear time with respect to the order of any tree T different from path. On the other hand, by Theorem 25 it can be noted that dim k (T ) < n. Therefore, according to Theorem 17, and considering that d(T ) = ς(T ) by Theorem 13, we deduce the next upper bound on k-partition dimension of T for any k ∈ {1, .…”

Section: The Particular Case Of Treesmentioning

confidence: 99%

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On the k-partition dimension of graphs

Estrada‐Moreno

2020

Theoretical Computer Science

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As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G = (V, E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u, v ∈ V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S 1 , . . . , S k ∈ Π such that d(u, S i ) = d(v, S i ) for every i ∈ {1, . . . , k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k ∈ {1, . . . , r}.

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“…For a tree T different from a path, the value of parameter ς(T ) can be computed in linear time with respect to the order of T , as it was shown in [45].…”

Section: On K-partition Dimensional Graphsmentioning

confidence: 99%

Section: Theorem 16 [11]mentioning

confidence: 99%

Section: The Particular Case Of Treesmentioning

confidence: 99%

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On the k-partition dimension of graphs

Estrada‐Moreno

2020

Theoretical Computer Science

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“…After these two papers, lot of research papers have been published related to its theoretical properties and applications. e theoretical studies on metric dimension are highly contributed by several authors like, for instance, independent resolving sets [3], resolving dominating sets [4], strong resolving sets [5], local resolving sets [6], k-metric generators [7,8], simultaneous metric generators [9], resolving partitions [10], strong resolving partitions [11], and k-antiresolving sets [12]. e metric dimension has many applications in the digital world and in mathematical chemistry such as network verification [13], robot navigation, image processing and pattern recognition [14], mastermind game [15], chemistry [16,17,18], and pharmaceutical chemistry [18].…”

Section: Introduction and Preliminarymentioning

confidence: 99%

On Resolvability Parameters of Some Wheel-Related Graphs

Yang

Rafiullah

Siddiqui

et al. 2019

Journal of Chemistry

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Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance between the vertex w and edge e is given by de,w=mindw,u,dw,v, A vertex w distinguishes two edges e1, e2∈E if dw,e1≠dw,e2. A set S is said to be resolving if every pair of edges of G is distinguished by some vertices of S. A resolving set with minimum cardinality is the basis for G, and this cardinality is the edge metric dimension of G, denoted by edimG. It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.

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“…This graph parameter was studied further in a number of other papers including, for instance, [10][11][12][13][14][15][16][17][18][19]. Several variations of metric generators including resolving dominating sets [20], independent resolving sets [21], local metric sets [22], strong resolving sets [23], mixed metric dimension [24], and -metric dimension [25] have since been introduced and studied.…”

Section: Introductionmentioning

confidence: 99%

The Metric Dimension of Some Generalized Petersen Graphs

Shao

Sheikholeslami

et al. 2018

Discrete Dynamics in Nature and Society

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The distance ( , V) between two distinct vertices and V in a graph is the length of a shortest ( , V)-path in . For an ordered subset = { 1 , 2 , . . . , } of vertices and a vertex V in , the code of V with respect to is the ordered -tupleis a resolving set for if every two vertices of have distinct codes. The metric dimension of is the minimum cardinality of a resolving set of . In this paper, we first extend the results of the metric dimension of ( , 3) and ( , 4) and study bounds on the metric dimension of the families of the generalized Petersen graphs (2 , ) and (3 , ). The obtained results mean that these families of graphs have constant metric dimension.

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Computing the k -metric dimension of graphs

Cited by 28 publications

References 20 publications

On the k-partition dimension of graphs

On the k-partition dimension of graphs

On Resolvability Parameters of Some Wheel-Related Graphs

The Metric Dimension of Some Generalized Petersen Graphs

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