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# On Partition Dimension and Domination of Abid-Waheed 〖(AW)〗_r^4 Graph

## Authors

• Jalal Hatem Hussein Bayati Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq https://orcid.org/0000-0002-3070-2391
• Abid Mahboob Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan. https://orcid.org/0000-0002-8784-6206
• Muhammad Waheed Rasheed Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan. https://orcid.org/0000-0002-3936-7673
• Dur e Najaf Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan. https://orcid.org/0000-0003-2023-2776

## Keywords:

Abid-Waheed Graph, Domination, Independent Domination, Partition Dimension, Restrained Domination.

## Abstract

A graph denoted by H, which has a simple link between its vertices, possesses the set of vertices V(H) . Given a graph,  a set that is dominant, is a subset of vertex set  such that any vertex outside of  is close to at least one vertex inside of . The smallest size of  for the dominating set is known as the graph’s domination number. When a linked graph H has a vertex x and a subset  of the vertex set, the separation between x and S is given by. Pertaining to an ordered k-partition  of , the illustration of  in relation to Π is to be the k-vectorAbid-Waheed graph  is a simply connected graph which contains  vertices and  edges for all  and  In this paper, we studied some results on the domination number, independent and restrained domination number denoted by  respectively in the Abid-Waheed graphs  and the relation between domination number, independent and restrained domination number. Also, the objective of this paper is to generate a partition dimension of.

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