Fault-Tolerant Resolvability and Extremal Structures of Graphs

Raza, Hassan; Hayat, Sakander; Imran, Muhammad; Pan, Xiang-Feng

doi:10.3390/math7010078

Cited by 57 publications

(30 citation statements)

References 29 publications

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“…Liu et al [23] studied the faulttolerant metric dimension of wheel related graphs. Raza et al [29] studied the extremal structure of graphs with respect to the fault-tolerant metric dimension. For more applications, we refer the reader to [37].…”

Section: A Literature Backgroundmentioning

confidence: 99%

Fault-Tolerant Metric Dimension of Interconnection Networks

et al. 2020

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A fixed interconnection parallel architecture is characterized by a graph, with vertices corresponding to processing nodes and edges representing communication links. An ordered set R of nodes in a graph G is said to be a resolving set of G if every node in G is uniquely determined by its vector of distances to the nodes in R. Each node in R can be thought of as the site for a sonar or loran station, and each node location must be uniquely determined by its distances to the sites in R. A fault-tolerant resolving set R for which the failure of any single station at node location v in R leaves us with a set that still is a resolving set. The metric dimension (resp. fault-tolerant metric dimension) is the minimum cardinality of a resolving set (resp. fault-tolerant resolving set). In this paper, we study the metric and fault-tolerant dimension of certain families of interconnection networks. In particular, we focus on the fault-tolerant metric dimension of the butterfly, the Benes and a family of honeycomb derived networks called the silicate networks. Our main results assert that three aforementioned families of interconnection have an unbounded fault-tolerant resolvability structures. We achieve that by determining certain maximal and minimal results on their fault-tolerant metric dimension.

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Section: A Literature Backgroundmentioning

confidence: 99%

Fault-Tolerant Metric Dimension of Interconnection Networks

et al. 2020

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“…For a graph G, dim(G) = n(G) − 3 and d(G) = 2 if and only if G is(K s ∪ B r ) + K t (s, r ≥ 2, t ≥ 1), (K s ∪ B r ) + K t (s, t ≥ 2, r ≥ 1), G = (K s + K t ) + B r (s, t ≥ 2, r ≥ 1), C5 or one of the graphs in Figures 2 and 3.Proof. It holds byLemmas 10,11, 12 and 14. …”

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confidence: 88%

“…The concepts of resolving set of a graph was first introduced by Slater [1] in 1975 and independently by Harary and Melter [2] in 1976. The metric dimension of a graph has been widely studied and a large number of related concepts have been extended (see [3][4][5][6][7][8][9][10][11]). As a parameter of a graph, it has been applied to lots of practical problems, such as robot navigation [12], connected joins in graphs and combinatorial optimization [13], and pharmaceutical chemistry [14].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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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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“…This concept was later studied by many researchers with unique modifications; for reference, see [4][5][6][7][8]. Some of the recent results on metric dimension and its further variations are studied in Shao et al [9] and Raza et al [10][11][12][13]. Lemma 1.…”

Section: Introductionmentioning

confidence: 99%

Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

Raza

2020

Front. Phys.

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Let Ŵ = (V, E) be a connected graph. A vertex i ∈ V recognizes two elements (vertices or edges) j, k ∈ E ∩ V, if d Ŵ (i, j) = d Ŵ (i, k). A set S of vertices in a connected graph Ŵ is a mixed metric generator for Ŵ if every two distinct elements (vertices or edges) of Ŵ are recognized by some vertex of S. The smallest cardinality of a mixed metric generator for Ŵ is called the mixed metric dimension and is denoted by β m. In this paper, the mixed metric dimension of a generalized Petersen graph P(n, 2) is calculated. We established that a generalized Petersen graph P(n, 2) has a mixed metric dimension equivalent to 4 for n ≡ 0, 2(mod4), and, for n ≡ 1, 3(mod4), the mixed metric dimension is 5. We thus determine that each graph of the family of a generalized Petersen graph P(n, 2) has a constant mixed metric dimension.

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Fault-Tolerant Resolvability and Extremal Structures of Graphs

Cited by 57 publications

References 29 publications

Fault-Tolerant Metric Dimension of Interconnection Networks

Fault-Tolerant Metric Dimension of Interconnection Networks

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

Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

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