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.
Highlights
In 1975, Slater [1] introduced the concept of a resolving set and its minimality within the graph, known as the metric dimension
We show that Γ has a fault-tolerant resolving set of cardinality 10
A method is presented to calculate the upper bounds on the fault-tolerant metric dimension of graphs
Summary
In 1975, Slater [1] introduced the concept of a resolving set and its minimality within the graph, known as the metric dimension. Fault-tolerant resolving sets enhance the applicability of classical resolving sets in graphs This shows that the fault-tolerant metric dimension possesses applicative superiority over the metric dimension. The fault-tolerant metric dimension of certain interesting graphs possessing chemical importance was studied in [20]. Raza et al [21,22] considered certain rotationally symmetric convex polytopes and studied their fault-tolerant metric dimension and binary-locating dominating sets. Based on the importance of fault-tolerant resolvability from both mathematical and applicative perspectives as discussed above, it is natural to study the mathematical properties of fault-tolerant resolving sets in graphs. We study the fault-tolerant resolvability for three infinite families of regular graphs and show some upper and lower bounds on their fault-tolerant metric dimension
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
