The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees

Basma Mohamed; Mohamed Amin

doi:doi:10.11648/j.acm.20231201.12

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The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees

Basma Mohamed, Mohamed Amin

Published in Applied and Computational Mathematics (Volume 12, Issue 1)

Received: 28 October 2022 Accepted: 16 November 2022 Published: 16 March 2023

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Abstract

Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.

Published in	Applied and Computational Mathematics (Volume 12, Issue 1)
DOI	10.11648/j.acm.20231201.12
Page(s)	9-14
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Resolving Set, Metric Dimension, Tadpole Graph, Bistar Tree, Coconut Tree

References

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APA Style

Basma Mohamed, Mohamed Amin. (2023). The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees. Applied and Computational Mathematics, 12(1), 9-14. https://doi.org/10.11648/j.acm.20231201.12

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ACS Style

Basma Mohamed; Mohamed Amin. The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees. Appl. Comput. Math. 2023, 12(1), 9-14. doi: 10.11648/j.acm.20231201.12

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AMA Style

Basma Mohamed, Mohamed Amin. The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees. Appl Comput Math. 2023;12(1):9-14. doi: 10.11648/j.acm.20231201.12

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@article{10.11648/j.acm.20231201.12,
  author = {Basma Mohamed and Mohamed Amin},
  title = {The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {1},
  pages = {9-14},
  doi = {10.11648/j.acm.20231201.12},
  url = {https://doi.org/10.11648/j.acm.20231201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231201.12},
  abstract = {Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.},
 year = {2023}
}

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TY  - JOUR
T1  - The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees
AU  - Basma Mohamed
AU  - Mohamed Amin
Y1  - 2023/03/16
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231201.12
DO  - 10.11648/j.acm.20231201.12
T2  - Applied and Computational Mathematics
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JO  - Applied and Computational Mathematics
SP  - 9
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PB  - Science Publishing Group
SN  - 2328-5613
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AB  - Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.
VL  - 12
IS  - 1
ER  -

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Author Information

Basma Mohamed

Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Shebin El-Kom, Egypt
Mohamed Amin

Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Shebin El-Kom, Egypt

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APA Style

Basma Mohamed, Mohamed Amin. (2023). The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees. Applied and Computational Mathematics, 12(1), 9-14. https://doi.org/10.11648/j.acm.20231201.12

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ACS Style

Basma Mohamed; Mohamed Amin. The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees. Appl. Comput. Math. 2023, 12(1), 9-14. doi: 10.11648/j.acm.20231201.12

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AMA Style

Basma Mohamed, Mohamed Amin. The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees. Appl Comput Math. 2023;12(1):9-14. doi: 10.11648/j.acm.20231201.12

Copy | Download

@article{10.11648/j.acm.20231201.12,
  author = {Basma Mohamed and Mohamed Amin},
  title = {The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {1},
  pages = {9-14},
  doi = {10.11648/j.acm.20231201.12},
  url = {https://doi.org/10.11648/j.acm.20231201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231201.12},
  abstract = {Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees
AU  - Basma Mohamed
AU  - Mohamed Amin
Y1  - 2023/03/16
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231201.12
DO  - 10.11648/j.acm.20231201.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 9
EP  - 14
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231201.12
AB  - Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.
VL  - 12
IS  - 1
ER  -

Copy | Download