TY - JOUR
T1 - Fault-tolerant resolvability and extremal structures of graphs
AU - Raza, Hassan
AU - Hayat, Sakander
AU - Imran, Muhammad
AU - Pan, Xiang Feng
N1 - Funding Information:
Funding: This research was supported by the Startup Research Grant Program of Higher Education Commission (HEC) Pakistan under Project# 2285 and grant No. 21-2285/SRGP/R&D/HEC/2018 received by Sakander Hayat. Muhammad Imran was supported by the Start-up Research Grant 2016 of United Arab Emirates University, Al Ain, United Arab Emirates via Grant No. G00002233 and UPAR Grant of United Arab Emirates University via Grant No. G00002590. APC was covered by Hassan Raza who was funded by a Chinese Government Scholarship.
Publisher Copyright:
© 2019 by the author.
PY - 2019/1/14
Y1 - 2019/1/14
N2 - 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.
AB - 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.
KW - Anti-prism graphs
KW - Extended Petersen graphs
KW - Fault-tolerant resolving set
KW - Resolving set
KW - Squared cycle graphs
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U2 - 10.3390/math7010078
DO - 10.3390/math7010078
M3 - Article
AN - SCOPUS:85060240189
SN - 2227-7390
VL - 7
JO - Mathematics
JF - Mathematics
IS - 1
M1 - 78
ER -