On Fault-Tolerant Metric Dimension of Heptagonal Circular Ladder and Its Related Graphs

Abstract

AbstractMetric dimension, these days, is an active research topic in combinatorics and graph theory. Given an undirected graph \(\mathbb {F}\), the combinatorial notion known as the metric dimension of \(\mathbb {F}\), is the least number of landmark nodes required to recognize (resolve) every pair of distinct nodes in the graph \(\mathbb {F}\), entirely based on the distance in graphs. A set D of vertices in \(\mathbb {F}\) is called a fault-tolerant resolving set (FTRS) for \(\mathbb {F}\), if \(D-\{y\}\) is still the resolving set for all \(y\in V(\mathbb {F})\). The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of \(\mathbb {F}\). In this article, we study the FTMD for a heptagonal circular ladder (\(\Gamma _{n}\), exists in the literature) and its related graphs and that is found to be constant for each of these families of the plane graphs. We show that the fault-tolerant metric dimension of \(\Gamma _{n}\) and a graph \(\Sigma _{n}\) obtained from it, is four and set upper and lower bound for the fault-tolerant metric dimension of another family \(\Upsilon _{n}\) of the graph obtained from \(\Gamma _{n}\).KeywordsHeptagonal circular ladderFault-tolerant resolving setConnected graphMetric dimensionPlane graphMSC(2020)05C1205C90