Journal of Prime Research in Mathematics

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

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