On the fault-tolerant metric dimension of certain interconnection networks

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

doi:10.1007/s12190-018-01225-y

Cited by 57 publications

(30 citation statements)

References 22 publications

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“…Fault-tolerant metric dimension has been studied in various fields. Raza et al [30] studied applications of the faulttolerant metric dimension in certain direct interconnection architectures. Somasundari et al [35] studied the faulttolerant metric dimension of oxide interconnection networks.…”

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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“…Recently, Raza et al [36] studied the fault-tolerant metric dimension of hexagonal, honeycomb, and hex-derived networks. See [37] for a study of hexagonal and honeycomb networks.…”

Section: Discussionmentioning

confidence: 99%

“…(ii) Investigate the fault-tolerant metric dimension of strongly regular graphs, such as the square grid graphs and the triangular graphs. (iii) In view of Raza et al [36], study the fault-tolerant resolvability in other direct and multiplex interconnection networks, such as the butterfly and Benes networks.…”

Section: Discussionmentioning

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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“…The affine double light-ray cone configuration is in accordance with the fundamental theorem of Alexandrov-Ovchinnikova of relativity theory and its spectral flattening procedure of tomographic affinization by means of the axis PSO(1, 1, R) of central projective collineations to display the central quantum compass state existence of a quantum state that corresponds precisely to a phase space point. Nevertheless, quantum entanglement highlights the phenomenon of collapse of quantum state in a specially subtle way because it involves pairs of projective measurements on the output of attributed non-classical quantum channels with at least two degrees of freedom or components [34,38,39], (Fig. 4).…”

Section: Copernican Spherical Holonomy and Kepplerian Calibrated Contmentioning

confidence: 99%

Applications of metaplectic cohomology and global-local contact holonomy

Schempp

2020

J. Appl. Math. Comput.

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The purpose of the paper is to elucidate by global-dual contact holonomy why one might expect the Foucault spherical pendulum's spin echo-stabilized, symplectic swing-plane through the rotation axis of the spinning earth to follow a parallel vector field as time passes at a velocity depending on the latitude of the swivel's location. The spinor geometric foundations of the paradigmatic Foucault spherical pendulum experiment of global-local contact holonomy exploit Hopf-Rinow type minimizing geodesic trajectories which appear as inertial traces designed by swing-planes on the terrestrial circular floor panel which is situated beneath the covariant motion of the oscillating pendulum device and where the linear traces are designing diameters of the spin structure induced Poincaré conformal hyperbolic open disc model. The quantum field theoretical evidence of the existence and uniqueness of categorizing parallel vector fields around smooth loops follows in terms of the open-book foliation with Hilbert space structures placed on the covariant pages. Keywords Dynamic quantization and kinematic relativization • Hopf principal circle bundle • Non-local quantum information processing • Supplementary open-book foliation • Hilbert bundle • Spin structure • Metaplectic Lie group Mp (2, R) • The metaplectic coadjoint orbit model • Half-spinor Maslov index Mathematics Subject Classification 81S10 • 70G65 • 70G45 • 57R17 • 53C80 • 53C38 • 53C27 • 53C22 • 53C12 • 22E25 • 14H81 • 11E08 • 11G07 • 11R11 Dedicated to the fourth centenary of the Kepplerian third spinor law H 3 of planetary dynamics.

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On the fault-tolerant metric dimension of certain interconnection networks

Cited by 57 publications

References 22 publications

Fault-Tolerant Metric Dimension of Interconnection Networks

Fault-Tolerant Metric Dimension of Interconnection Networks

Fault-Tolerant Resolvability and Extremal Structures of Graphs

Applications of metaplectic cohomology and global-local contact holonomy

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