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

The comparative analysis of metric and edge metric dimension of some subdivisions of the wheel graph

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500625 ◽

2020 ◽

pp. 2150062

Author(s):

Zahid Raza ◽

M. S. Bataineh

Keyword(s):

Comparative Analysis ◽

Metric Dimension ◽

Wheel Graph ◽

Wheel Graphs ◽

Metric Dimensions ◽

Appl Math

The aim of this study is to compute the edge metric dimension of some subdivision of the wheel graphs. In particular, we determine and compare the metric and edge metric dimensions of the graphs obtained after the cycle, spoke and barycentric subdivisions of the wheel graph. Furthermore, some families of graphs have been constructed through subdivision process for which [Formula: see text], and also [Formula: see text] which partially answer a question in [A. Kelenc, N. Tratnik and I. G. Yero, Uniquely identifying the edges of a graph: The edge metric dimension, Discrete Appl. Math. 251 (2018) 204–220].

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On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.4 ◽

2018 ◽

Vol 2 (2) ◽

pp. 88

Author(s):

Rokhana Ayu Solekhah ◽

Tri Atmojo Kusmayadi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Minimum Cardinality ◽

Wheel Graph ◽

Metric Set ◽

Metric Basis ◽

Fan Graph

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w1), d(v, w2), ..., d(v, wn)), where d(v, wi) is the distance between v and wi for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a t-fold wheel graph, Pn ⊙ Km graph, and generalized fan graph.

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The Vertex-Edge Resolvability of Some Wheel-Related Graphs

Journal of Mathematics ◽

10.1155/2021/1859714 ◽

2021 ◽

Vol 2021 ◽

pp. 1-16

Author(s):

Bao-Hua Xing ◽

Sunny Kumar Sharma ◽

Vijay Kumar Bhat ◽

Hassan Raza ◽

Jia-Bao Liu

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Wheel Graph ◽

Wheel Graphs ◽

Metric Dimensions ◽

Mixed Metric

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

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ON THE STRONG METRIC DIMENSION OF SOME RELATED WHEEL GRAPH

Far East Journal of Mathematical Sciences (FJMS) ◽

10.17654/ms099091325 ◽

2016 ◽

Vol 99 (9) ◽

pp. 1325-1334 ◽

Cited By ~ 3

Author(s):

Tri Atmojo Kusmayadi ◽

Sri Kuntari ◽

Deddy Rahmadi ◽

Fithri Annisatun Lathifah

Keyword(s):

Metric Dimension ◽

Wheel Graph ◽

Strong Metric Dimension

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Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes

Mathematical Problems in Engineering ◽

10.1155/2020/1216542 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Zhi-Bo Zheng ◽

Ashfaq Ahmad ◽

Zaffar Hussain ◽

Mobeen Munir ◽

Muhammad Imran Qureshi ◽

...

Keyword(s):

Present Article ◽

Open Problem ◽

Ordered Set ◽

Fault Tolerant ◽

Convex Polytopes ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Wheel Graph

For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.

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Power graphs and exchange property for resolving sets

Open Mathematics ◽

10.1515/math-2019-0093 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1303-1309 ◽

Cited By ~ 1

Author(s):

Ghulam Abbas ◽

Usman Ali ◽

Mobeen Munir ◽

Syed Ahtsham Ul Haq Bokhary ◽

Shin Min Kang

Keyword(s):

Present Article ◽

Linear Algebra ◽

Finite Groups ◽

Robot Navigation ◽

Simple Graph ◽

Exchange Property ◽

Metric Dimension ◽

Sufficient Condition ◽

Resolving Set ◽

Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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Metric Dimension Parameterized By Treewidth

Algorithmica ◽

10.1007/s00453-021-00808-9 ◽

2021 ◽

Author(s):

Édouard Bonnet ◽

Nidhi Purohit

Keyword(s):

Computable Function ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Minimum Size ◽

Metric Dimension ◽

Resolving Set ◽

Input Graph ◽

Outerplanar Graphs ◽

Bounded Treewidth ◽

Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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