Metric Dimension and Some Related Parameters of Different Classes of Benzenoid System

Muhammad Imran Qureshi\(^a\), Zill e Shams \(^{b,∗}\), Rukhsar Zireen\(^a\), Sana Saeed\(^a\)

\(^a\)Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan.

\(^b\)Department of Mathematics, The Women University Multan.

Correspondence should be addressed to: imranqureshi18@gmail.com,shabbir.khurram@gmail.com,rukhsarzireen7@gmail.com,sanasaeed181soudren@gmail.com

Copyright © 2024 Muhammad Imran Qureshi, Zill e Shams, Rukhsar Zireen, Sana Saeed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 16 April 2024 Accepted: 3 July 2024; Published Online: 05 July 2024

Abstract

The resolving set for connected graphs has become one of the most important concept due to its applicability in networking, robotics and computer sciences. Let G be a simple and connected graph, an ordered-subset B of V (G) is called resolving set of G, if every distinct vertex of G have different metric code w.r.t B. Smallest resolving set of G is known as basis of G and size of basis set is called as metric dimension(MD) of graph G. A resolving set B′ of G is known as fault-tolerant resolving set(FTRS), ifB′\{v} is also resolving set, ∀ v ϵ B′. Such set B′ with smallest size is termed as fault-tolerant metric basis and the cardinality of this set is called fault-tolerant metric dimension(FTMD) of graph G. A FTMD set B′ for which the system failure at vertex location v of any station still provide us a resolving set. In this article, we have provided the MD and FTMD for triangular benzenoid system and hourglass benzenoid system.

Keywords:

Metric dimension, Resolving set, FTMD, Triangular benzenoid system, Hourglass benzenoid system