The k-Metric Dimension of Corona Product Graphs

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

doi:10.1007/s40840-015-0282-2

Cited by 44 publications

(51 citation statements)

References 14 publications

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“…Our next result on graphs of diameter grater than or equal to six, is a direct consequence of Theorem 26. A fan graph is defined as the join graph K 1 + P n , where P n is a path of order n, and a wheel graph is defined as the join graph K 1 + C n , where C n is a cycle graph of order n. The following closed formulae for the k-metric dimension of fan and wheel graphs were obtained in [4,10]. Since these graphs have diameter two, we express the result in terms of the k-adjacency dimension.…”

Section: The K-adjacency Dimension Of Join Graphsmentioning

confidence: 99%

“…The concept of k-metric generator introduced by Estrada-Moreno, Yero and Rodríguez-Velázquez [4,6], is a natural extension of the concept of metric generator. A set S ⊆ V is said to be a k-metric generator for G if and only if any pair of vertices of G is distinguished by at least k elements of S, i.e., for any pair of different vertices u, v ∈ V , there exist at least k vertices w 1 , w 2 , ..., w k ∈ S such that d G (u, w i ) = d G (v, w i ), for every i ∈ {1, ..., k}.…”

Section: Introductionmentioning

confidence: 99%

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On the adjacency dimension of graphs

Estrada‐Moreno

Ramírez-Cruz

Rodríguez-Velázquez

2016

APPL ANAL DISCRETE M

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A generator of a metric space is a set S of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of S. Given a simple graph G = (V, E), we define the distance function dG,2 : V × V → N ∪ {0}, as dG,2(x, y) = min{dG(x, y), 2}, where dG(x, y) is the length of a shortest path between x and y and N is the set of positive integers. Then (V, dG,2) is a metric space. We say that a set S ⊆ V is a k-adjacency generator for G if for every two vertices x, y ∈ V , there exist at least k vertices w1, w2, ..., w k ∈ S such that dG,2(x, wi) = dG,2(y, wi), for every i ∈ {1, ..., k}.A minimum cardinality k-adjacency generator is called a k-adjacency basis of G and its cardinality, the k-adjacency dimension of G.In this article we study the problem of finding the k-adjacency dimension of a graph. We give some necessary and sufficient conditions for the existence of a k-adjacency basis of an arbitrary graph G and we obtain general results on the k-adjacency dimension, including general bounds and closed formulae for some families of graphs. In particular, we obtain closed formulae for the k-adjacency dimension of join graphs G + H in terms of the k-adjacency dimension of G and H. These results concern the k-metric dimension, as join graphs have diameter two. As we can expect, the obtained results will become important tools for the study of the k-metric dimension of lexicographic product graphs and corona product graphs. Moreover, several results obtained in this paper need not be restricted to the metric dG,2, they can be expressed in a more general setting, for instance, by using the metric dG,t(x, y) = min{dG(x, y), t} for t ∈ N.

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Section: The K-adjacency Dimension Of Join Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the adjacency dimension of graphs

Estrada‐Moreno

Ramírez-Cruz

Rodríguez-Velázquez

2016

APPL ANAL DISCRETE M

Self Cite

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show abstract

“…In 2015 Estrada-Moreno et al [1] have found themetric dimension in the path graph, cycle graph, tree graph, and join operation between two graphs. Then in 2016 Estrada-Moreno et al [2] have found -metric dimension on the corona operation between two graphs. In 2017 Geetha and Sooryanarayana [9] have found the -metric dimension in the graph of cartesian product operation results.…”

Section: Introductionmentioning

confidence: 99%

The k-Metric Dimension of Nk + Pn Graph and Starbarbell Graph

Saidah¹,

Kusmayadi²

2020

Proceedings of the SEMANTIK Conference of Mathematics Education (SEMANTIK 2019)

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Let be a simple connected graph with a set of vertices ( ) and set of edges ( ). The distance between two vertices and in a graph are the shortest path length between two vertices and denoted by ( , ). Let be a positive integer, ⊆ with is a -metric generator if and only if for each different vertex pair , ∈ there are at least vertices 1 , 2 , … , ∈ and fulfill ( , ) ≠ ( , ) with ∈ {1, 2, … , }. Minimum cardinality of a -metric generator of a graph is called the basis -metric of graph . The number of elements on the basis of -metric graph are called -metric dimension of graph and denoted by ( ). + is the result of a join operation between null graph and path graph with , ≥ 2. Starbarbell graph denoted by 1, 2,…, is a graph formed from a star graph 1, and complete graph then merge one vertex from each with ℎ leaf of 1, with ≥ 3, 1 ≤ ≤ , and ≥ 2. In this paper, we determine the -metric dimension of + graph and starbarbell graph.

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“…In 2015, Estrada-Moreno et al [4] discovered the k-metric dimension of path graphs, cycle graphs, tree graphs and graphs resulting from joint operations with vertices on each graph are twin vertices. In 2016, Estrada-Moreno et al [5] discovered the kmetric dimension of corona product graphs. In 2017, Geetha and Sooryanarayana [6] discovered the 2-metric dimension of Cartesian product graphs.…”

Section: Introductionmentioning

confidence: 99%

On the k-Metric Dimension of a Barbell Graph and a t-fold Wheel Graph

Setyawan¹,

Kusmayadi²

2020

Proceedings of the SEMANTIK Conference of Mathematics Education (SEMANTIK 2019)

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Let G be a connected and simple graph with the vertex set V(G) and the edge set E(G). The set S ⊆ V (G) is called a k-metric generator for G if and only if for every two pairs different vertices u,v ∈ V(G), there are at least k vertices w1,w2, . . .,wk ∈ S such that d(u,wi) ≠ d(v,wi) for every i ∈ {1, 2, …, k}, with d(u,v) is the length of shortest uv path. A minimum k-metric generator is called a k-metric basis and its cardinality is called the k-metric dimension of G, denoted by dimk(G). A barbell graph Bn,n for n ≥ 3 is the simple graph obtained from two complete graph Kn connected by a bridge. A t-fold wheel graph Wnt for t ≥ 2 and n ≥ 3 is the simple graph that contain the central t vertex which are adjacent to each vertex in a cycle, but not adjacent to each other. In this paper, we determine the kmetric dimension of a barbell graph and a t-fold wheel graph.

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The k-Metric Dimension of Corona Product Graphs

Cited by 44 publications

References 14 publications

On the adjacency dimension of graphs

On the adjacency dimension of graphs

The k-Metric Dimension of Nk + Pn Graph and Starbarbell Graph

On the k-Metric Dimension of a Barbell Graph and a t-fold Wheel Graph

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