Fault-Tolerant Resolvability in Some Classes of Line Graphs

Guo, Xuan; Faheem, Muhammad; Zahid, Zohaib; Nazeer, Waqas; Li, Jingjng

doi:10.1155/2020/1436872

Cited by 6 publications

(3 citation statements)

References 11 publications

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“…Laxman in [22] computed the lower bound of the FTMD for the cube of the path graph. Recently, the FTMD for the line graphs was studied by Guo et al in [23], and they computed it for the line graphs of the prism and necklace graphs.…”

Section: Introductionmentioning

confidence: 99%

Fault‐Tolerant Resolvability in Some Classes of Subdivision Graphs

Faheem

Zahid

Alrowaili

et al. 2022

Journal of Mathematics

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The concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure. RS has been used as a sensor in an indoor positioning system to find an interrupter. Many terminologies in machine learning have also been used to diagnose the interrupter in the systems of marine and gas turbines using sensory data. We proposed a fault-tolerant self-stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails. If the elimination of any element from a RS is still a RS, then the RS is considered as a fault-tolerant resolving set (FTRS), and the fault-tolerant metric dimension (FTMD) is its minimum cardinality. In this paper, we calculated the FTMD of the subdivision graphs of the necklace and prism graphs. We also found that this invariant has constant values for both graphs.

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Section: Introductionmentioning

confidence: 99%

Fault‐Tolerant Resolvability in Some Classes of Subdivision Graphs

Faheem

Zahid

Alrowaili

et al. 2022

Journal of Mathematics

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“…Voronov [26] and Raza et al [27] determined some important upper bounds for the king's and extended Petersen graphs, respectively. Guo et al [28] computed the FTMD for the line graphs of the families of necklace and prism graphs. Faheem et al [29] calculated this invariant for the subdivision graphs of the same families of graphs.…”

Section: Introductionmentioning

confidence: 99%

On the FAULT-TOLERANT Resolvability in Line Graphs of Dragon and Kayak Paddles Graphs

Faheem

Zahid

2022

Mathematical Problems in Engineering

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Because of its wide range of applications, metric resolvability has been used in chemical structures, computer networks, and electrical circuits. It has been applied as a node (sensor) in an electric circuit. The electric circuit will not be able to flow current if one node (sensor) in that chain becomes faulty. The fault-tolerant selfstable circuit is a circuit that permits the current flow even if one of the nodes (sensors) becomes faulty. If the removal of any node from a resolving set (RS) of the circuit is still a RS, then the RS of the circuit is considered a fault-tolerant resolving set (FTRS) and the fault-tolerant metric dimension (FTMD) is its minimum cardinality. Even though the problem of finding the exact values of MD in line graphs seems to be even harder, the FTMD for the line graphs was first discussed by Guo et al. [13]. Ahmad et al. [5] determined the precise value of the MD for the line graph of the kayak paddle graph. We calculate the precise value of the FTMD for the line graph in this family of graphs. The FTMD is a more generalized invariant than the MD. We also consider the problem of obtaining a precise value for this parameter in the line graph of the dragon graph. It is concluded that these families have a constant FTMD.

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“…Mithun et al [24] considered FTMD for the class of circulant graph C n (1, 2, 3). For, more in-depth review of this particular topic we refer to some of the recent results in [15,17,20,25].…”

Section: Introductionmentioning

confidence: 99%

Rotationally symmetrical plane graphs and their Fault-tolerant metric dimension

Sharma¹,

Bhat

2021

AUCMCS

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Consider a robot which is investigating in a space exhibited by a graph (network), and which needs to know its current location. It can grant a sign to find how far it is from each among a lot of fixed places of interest (tourist spots or landmarks). We study the problem of calculating the minimum number of tourist spots required, and where they ought to be set, with the ultimate objective that the robot can generally decide its location. The set of nodes where the places of interest are placed is known as the metric basis of the graph, and the cardinality of tourist spots is known as the location number (or metric dimension) of the graph. Another graph invariant related to resolving set (say $\mathfrak{L}$) is the fault-tolerant resolving set $\mathfrak{L}^{\ast}$, in which the expulsion of a discretionary vertex from $\mathfrak{L}$ keeps up the resolvability. The problem of characterizing the classes of plane graphs with a bounded fault-tolerant metric dimension is of great interest nowadays. In this article, we obtain the fault-tolerant metric dimension of three interminable classes of symmetrical plane graphs, that are found to be constant for each of these three families of the plane graphs. We set lower and upper bounds for the fault-tolerant metric dimension of these three classes of the plane graphs.

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Fault-Tolerant Resolvability in Some Classes of Line Graphs

Cited by 6 publications

References 11 publications

Fault‐Tolerant Resolvability in Some Classes of Subdivision Graphs

Fault‐Tolerant Resolvability in Some Classes of Subdivision Graphs

On the FAULT-TOLERANT Resolvability in Line Graphs of Dragon and Kayak Paddles Graphs

Rotationally symmetrical plane graphs and their Fault-tolerant metric dimension

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