Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes

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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Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

Symmetry ◽

10.3390/sym10080300 ◽

2018 ◽

Vol 10 (8) ◽

pp. 300 ◽

Cited By ~ 5

Author(s):

Zafar Hussain ◽

Mobeen Munir ◽

Maqbool Chaudhary ◽

Shin Kang

Keyword(s):

Electronic Structure ◽

Transport Properties ◽

Present Article ◽

Connected Graph ◽

Metric Dimension ◽

Lattice Structures ◽

Resolving Set ◽

Minimum Cardinality ◽

Mathematical Sciences ◽

Metric Basis

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

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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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Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽

10.3390/math10010124 ◽

2022 ◽

Vol 10 (1) ◽

pp. 124

Author(s):

Laxman Saha ◽

Rupen Lama ◽

Kalishankar Tiwary ◽

Kinkar Chandra Das ◽

Yilun Shang

Keyword(s):

Connected Graph ◽

Fault Tolerant ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Minimum Cardinality ◽

Vertex Set

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

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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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Non-Isolated Resolving Number of Graph with Pendant Edges

Journal of Interconnection Networks ◽

10.1142/s0219265919500038 ◽

2019 ◽

Vol 19 (02) ◽

pp. 1950003

Author(s):

RIDHO ALFARISI ◽

DAFIK ◽

ARIKA INDAH KRISTIANA ◽

IKA HESTI AGUSTIN

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Isolated Vertex

We consider V, E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = {w1, w2, w3, …, wk} of vertices and a vertex v ∈ G, the ordered r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)) of k-vector is representations of v with respect to W, where d(v, w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of |W|. Furthermore, the resolving set W of graph G is called non-isolated resolving set if there is no ∀v ∈ W induced by non-isolated vertex. While a non-isolated resolving number, denoted by nr(G), is the minimum cardinality of non-isolated resolving set in graph. In this paper, we study the non isolated resolving number of graph with any pendant edges.

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FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1904781s ◽

2019 ◽

pp. 781

Author(s):

Narjes Seyedi ◽

Hamid Reza Maimani

Keyword(s):

Fault Tolerant ◽

Additive Group ◽

Metric Dimension ◽

Circulant Graph ◽

Resolving Set ◽

Circulant Graphs ◽

Minimum Cardinality ◽

Vertex Set

A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(G)$. A resolving set $W'$ for $G$ is fault-tolerant if $W'\setminus \left\lbrace w\right\rbrace $ for each $w$ in $W'$, is also a resolving set and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set, denoted by $\beta'(G)$. The circulant graph is a graph with vertex set $\mathbb{Z}_{n}$, an additive group of integers modulo $n$, and two vertices labeled $i$ and $j$ adjacent if and only if $i -j \left( mod \ n \right) \in C$, where $C \in \mathbb{Z}_{n}$ has the property that $C=-C$ and $0 \notin C$. The circulant graph is denoted by $X_{n,\bigtriangleup}$ where $\bigtriangleup = \vert C\vert$. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs $X_{n,3}$ with connection set $C=\lbrace 1,\dfrac{n}{2},n-1\rbrace$ and circulant graphs $X_{n,4}$ with connection set $C=\lbrace \pm 1,\pm 2\rbrace$.

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Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

Mathematics ◽

10.3390/math7050479 ◽

2019 ◽

Vol 7 (5) ◽

pp. 479 ◽

Cited By ~ 1

Author(s):

Juan Wang ◽

Lianying Miao ◽

Yunlong Liu

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Distance Matrix ◽

Metric Dimension ◽

Extremal Graphs ◽

Resolving Set ◽

Minimum Cardinality

Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

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On The Local Metric Dimension of Line Graph of Special Graph

CAUCHY ◽

10.18860/ca.v4i3.3694 ◽

2016 ◽

Vol 4 (3) ◽

pp. 125

Author(s):

Marsidi Marsidi ◽

Dafik Dafik ◽

Ika Hesti Agustin ◽

Ridho Alfarisi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Line Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality

Let G be a simple, nontrivial, and connected graph. is a representation of an ordered set of k distinct vertices in a nontrivial connected graph G. The metric code of a vertex v, where , the ordered of k-vector is representations of v with respect to W, where is the distance between the vertices v and wi for 1≤ i ≤k. Furthermore, the set W is called a local resolving set of G if for every pair u,v of adjacent vertices of G. The local metric dimension ldim(G) is minimum cardinality of W. The local metric dimension exists for every nontrivial connected graph G. In this paper, we study the local metric dimension of line graph of special graphs , namely path, cycle, generalized star, and wheel. The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common.

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On the joint product graphs with respect to dominant metric dimension

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500105 ◽

2020 ◽

pp. 2150010

Author(s):

Liliek Susilowati ◽

Imroatus Sa’adah ◽

Utami Dyah Purwati

Keyword(s):

Graph Theory ◽

Connected Graph ◽

Ordered Set ◽

Dominating Set ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Joint Product ◽

Product Graphs

Some concepts in graph theory are resolving set, dominating set, and dominant metric dimension. A resolving set of a connected graph [Formula: see text] is the ordered set [Formula: see text] such that every pair of two vertices [Formula: see text] has the different representation with respect to [Formula: see text]. A Dominating set of [Formula: see text] is the subset [Formula: see text] such that for every vertex [Formula: see text] in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. A dominant resolving set of [Formula: see text] is an ordered set [Formula: see text] such that [Formula: see text] is a resolving set and a dominating set of [Formula: see text]. The minimum cardinality of a dominant resolving set is called a dominant metric dimension of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the dominant metric dimension of the joint product graphs.

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Fault-Tolerant Resolvability in Some Classes of Line Graphs

Mathematical Problems in Engineering ◽

10.1155/2020/1436872 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Xuan Guo ◽

Muhammad Faheem ◽

Zohaib Zahid ◽

Waqas Nazeer ◽

Jingjng Li

Keyword(s):

Graph Theory ◽

Fault Tolerance ◽

Fault Tolerant ◽

Stable System ◽

Subject Classification ◽

Metric Dimension ◽

Line Graphs ◽

Resolving Set ◽

Minimum Cardinality

Fault tolerance is the characteristic of a system that permits it to carry on its intended operations in case of the failure of one of its units. Such a system is known as the fault-tolerant self-stable system. In graph theory, if we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this paper, we determine the fault-tolerant resolvability in line graphs. As a main result, we computed the fault-tolerant metric dimension of line graphs of necklace and prism graphs (2010 Mathematics Subject Classification: 05C78).

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