On the fault-tolerant metric dimension of convex polytopes

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

doi:10.1016/j.amc.2018.07.010

Cited by 61 publications

(34 citation statements)

References 20 publications

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“…Krishnan et al [21] studied applications of the faulttolerant metric dimension in crystalline structures. Raza et al [31] studied applications of fault-tolerant metric dimension in convex polytopes. Liu et al [23] studied the faulttolerant metric dimension of wheel related graphs.…”

Section: A Literature Backgroundmentioning

confidence: 99%

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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%

“…For any set W ⊂ V (G), let γ(W ) be the set of all the common neighbors of vertices in W . Based on this concept, Raza et al [31] determined the following interesting relation between a resolving and a fault-tolerant resolving set.…”

Section: Theorem 13 [18]mentioning

confidence: 99%

Fault-Tolerant Metric Dimension of Interconnection Networks

et al. 2020

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“…It uses R in a graph to produce a fault-tolerant resolving set within it. In view of Lemma 2, for a given resolving set R of a graph Γ, finding the set R to evaluate the corresponding fault-tolerant resolving set seems tedious due to the calculation of the set T(w) for a vertex w ∈ R. Raza et al [21] further simplify this lemma so that one does not have to check every vertex x of Γ to verify whether or not it belongs to T(w) for some w ∈ R. Now, for vertices x and y in Γ, we let λ(x, y) be a set of common neighbors of these vertices and, for some Q ⊂ V(Γ), let λ(Q) be the set of common neighbors of each vertex in Q. The following lemma is a key result for finding upper bounds on the fault-tolerant metric dimension of a given graph.…”

Section: Relation Between Resolving Sets and Fault-tolerant Resolvingmentioning

confidence: 99%

“…The fault-tolerant metric dimension of certain interesting graphs possessing chemical importance was studied in [20]. Recently, Raza et al [21,22] considered certain rotationally symmetric convex polytopes and studied their fault-tolerant metric dimension and binary-locating dominating sets. The reader is referred to [23] for consideration of fault-tolerant resolvability as an optimization problem and its applicative perspective.…”

Section: Introductionmentioning

confidence: 99%

Fault-Tolerant Resolvability and Extremal Structures of Graphs

et al. 2019

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In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

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“…2Study the minimum metric dimension problem for G. This problem is studied in [17][18][19]32,34] for other families of regular and non-regular convex polytopes. 3Study fault-tolerant resolvability of G. A similar study for other classes of convex polytopes is conducted by Raza et al [35] and Salman et al [22].…”

Section: Constructionmentioning

confidence: 99%

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

2018

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A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.

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On the fault-tolerant metric dimension of convex polytopes

Cited by 61 publications

References 20 publications

Fault-Tolerant Metric Dimension of Interconnection Networks

Fault-Tolerant Metric Dimension of Interconnection Networks

Fault-Tolerant Resolvability and Extremal Structures of Graphs

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

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