Fault-Tolerant Resolvability and Extremal Structures of Graphs

Hassan Raza, Sakander Hayat, Muhammad Imran and Xiang-Feng Pan
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Hassan Raza: School of Mathematical Sciences, Anhui University, Hefei 230601, China
Sakander Hayat: Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Swabi 23460, Pakistan
Muhammad Imran: Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, UAE
Xiang-Feng Pan: School of Mathematical Sciences, Anhui University, Hefei 230601, China

Mathematics, 2019, vol. 7, issue 1, 1-19

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.

Keywords: resolving set; fault-tolerant resolving set; extended Petersen graphs; anti-prism graphs; squared cycle graphs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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