Computing vertex resolvability of benzenoid tripod structure

Maryam Salem Alatawi; Ali Ahmad; Ali N. A. Koam; Sadia Husain; Muhammad Azeem; Maryam Salem Alatawi; Ali Ahmad; Ali N. A. Koam; Sadia Husain; Muhammad Azeem

doi:10.3934/math.2022387

AIMS Mathematics

2022, Volume 7, Issue 4: 6971-6983. doi: 10.3934/math.2022387

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Research article

Computing vertex resolvability of benzenoid tripod structure

1.
Department of Mathematics Faculty of Sciences, University of Tabuk 71491 Tabuk, Saudi Arabia
2.
College of Computer Science and Information Technology, Jazan University, Jazan, Saudi Arabia
3.
Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi Arabia
4.
Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, Pakistan

Received: 15 September 2021 Revised: 22 January 2022 Accepted: 25 January 2022 Published: 07 February 2022
MSC : 05C12, 05C09, 05C92

In this paper, we determine the exact metric and fault-tolerant metric dimension of the benzenoid tripod structure. We also computed the generalized version of this parameter and proved that all the parameters are constant. Resolving set $ {L} $ is an ordered subset of nodes of a graph $ {C} $, in which each vertex of $ {C} $ is distinctively determined by its distance vector to the nodes in $ {L} $. The cardinality of a minimum resolving set is called the metric dimension of $ {C} $. A resolving set $ L_{f} $ of $ {C} $ is fault-tolerant if $ {L}_{f}\setminus{b} $ is also a resolving set, for every $ {b} $ in $ {L}_{f}. $ Resolving set allows to obtain a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represents the atom and bond types, respectively.
- node-resolvability,
- fault-tolerant node-resolvability,
- benzenoid structure,
- benzenoid tripod
Citation: Maryam Salem Alatawi, Ali Ahmad, Ali N. A. Koam, Sadia Husain, Muhammad Azeem. Computing vertex resolvability of benzenoid tripod structure[J]. AIMS Mathematics, 2022, 7(4): 6971-6983. doi: 10.3934/math.2022387

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Abstract

In this paper, we determine the exact metric and fault-tolerant metric dimension of the benzenoid tripod structure. We also computed the generalized version of this parameter and proved that all the parameters are constant. Resolving set $ {L} $ is an ordered subset of nodes of a graph $ {C} $, in which each vertex of $ {C} $ is distinctively determined by its distance vector to the nodes in $ {L} $. The cardinality of a minimum resolving set is called the metric dimension of $ {C} $. A resolving set $ L_{f} $ of $ {C} $ is fault-tolerant if $ {L}_{f}\setminus{b} $ is also a resolving set, for every $ {b} $ in $ {L}_{f}. $ Resolving set allows to obtain a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represents the atom and bond types, respectively.

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