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Mathematics 2019, 7(1), 78; https://doi.org/10.3390/math7010078

Fault-Tolerant Resolvability and Extremal Structures of Graphs

1
School of Mathematical Sciences, Anhui University, Hefei 230601, China
2
Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Swabi 23460, Pakistan
3
Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, UAE
4
School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan
*
Author to whom correspondence should be addressed.
Received: 25 November 2018 / Revised: 31 December 2018 / Accepted: 10 January 2019 / Published: 14 January 2019
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Abstract

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. View Full-Text
Keywords: resolving set; fault-tolerant resolving set; extended Petersen graphs; anti-prism graphs; squared cycle graphs resolving set; fault-tolerant resolving set; extended Petersen graphs; anti-prism graphs; squared cycle graphs
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Raza, H.; Hayat, S.; Imran, M.; Pan, X.-F. Fault-Tolerant Resolvability and Extremal Structures of Graphs. Mathematics 2019, 7, 78.

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