Domination Number and Secure Resolving Sets in Cyclic Networks

Basma Mohamed; Mohamed Amin

doi:doi:10.11648/j.acm.20231202.12

| Peer-Reviewed

Domination Number and Secure Resolving Sets in Cyclic Networks

Basma Mohamed, Mohamed Amin

Published in Applied and Computational Mathematics (Volume 12, Issue 2)

Received: 21 April 2023 Accepted: 11 May 2023 Published: 29 May 2023

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Consider a robot that is navigating a graph-based space and is attempting to determine where it is right now. To determine how distant it is from each group of fixed landmarks, it can send a signal. We discuss the problem of determining the minimum number of landmarks necessary and their optimal placement to ensure that the robot can always locate itself. The number of landmarks is referred to as the graph's metric dimension, and the set of nodes on which they are distributed is known as the graph's metric basis. On the other hand, the metric dimension of a graph G is the minimum size of a set w of vertices that can identify each vertex pair of G by the shortest-path distance to a particular vertex in w. It is an NP-complete problem to determine the metric dimension for any network. The metric dimension is also used in a variety of applications, including geographic routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we investigate the exact value of the secure resolving set of some networks, such as trapezoid network, Z-(P_n) network, open ladder network, tortoise network and network. We also determine the domination number of the networks, such as the twig network T_m, double fan network F_2,n, bistar network B_n,n and linear kc₄ – snake network.

Published in	Applied and Computational Mathematics (Volume 12, Issue 2)
DOI	10.11648/j.acm.20231202.12
Page(s)	42-45
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

Domination Number, Secure Resolving Set, Twig Graph and Linear kc4 - Snake Graph

References

[1]	P. J. Slater, “Leaves of trees”, Proc. 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, (1975), pp. 549–559.
[2]	F. Harary, R. A. Melter, “On the metric dimension of a graph”, Ars Combinatoria, 2 (1976), pp. 191–195.
[3]	E. J. Cockayne, O. Favaron, C. M. Mynhardt, “Secure domination, weak roman domination and forbidden subgraph”, Bull. Inst. Comb. Appl., 39 (2003), pp. 87–100.
[4]	H. Subramanian and S. Arasappan, “Secure Resolving Sets in a Graph”, Symmetry, 10 (10) 2018.
[5]	C. Berge, “Theory of Graphs and its Applications”, Methuen, London, 1962.
[6]	O. Ore, “Theory of graphs”, American Mathematical Society Translations, 38 (1962), pp. 206–212.
[7]	J. L. Hurink and T. Nieberg, “Approximating minimum independent dominating sets in wireless networks”, Information processing letters, 109 (2008), pp. 155-160.
[8]	A. A. Khalil, “Determination and testing the Domination Numbers of Helm Graph, Web graph and Levi Graph Using MATLAB”, Science Education, 24 (2) 2011.
[9]	C. S. Nagabhushana, B. N. Kavitha and H. M. Chudamani, “Split and Equitable Domination of Some Special Graph”, International Journal of Science Technology & Engineering, 4 (2) 2017.
[10]	K. B. Murthy, “The End equitable Domination of Dragon and some Related Graphs”, Computer and Mathematical sciences, 7 (3) 2016, pp. 160-167.
[11]	B. N. Kavitha and I. Kelkar, “Split and Equitable domination in book graph and stacked book graph”, International Journal of Advanced Research in Computer Science, 8 (6) 2017.
[12]	A. Sugumaran and E. Jayachandran, “Domination number of some graphs”, International Scientific Development and Research, 3 (11) 2018, pp. 386 - 391.
[13]	G. B. Mertzios and D. G. Corneil, “Vertex splitting and the recognition of trapezoid graphs”, Discrete Applied Mathematics, 159 (11) 2011, pp. 1131-1147.
[14]	A. N. Murugan and G. Esther, “Path Related Mean Cordial Graphs”, Global Research in Mathematical Archives, 2 (3) 2014, pp. 74-85.
[15]	P. Sumathi, A. Rathi and A. Mahalakshmi, “Quotient labeling of corona of ladder graphs”, International Journal of Innovative Research in Applied Sciences and Engineering, 1 (3) 2017, pp. 80-85.
[16]	N. Murugesan and R. Uma, “Super vertex gracefulness of some special graphs”, Mathematics, 11 (3) 2015, pp. 7-15.
[17]	A. H. Rokad, “Product cordial labeling of double wheel and double fan related graphs”, Kragujevac Journal of Mathematics, 43 (1) 2019, pp. 7–13.
[18]	S. K. Vaidya and N. H. Shah, “On square divisor cordial graphs”, Scientific Research, 6 (3) 2014, pp. 445-455.
[19]	B. Gayathri, R. Thayalarajan, “Square difference mean labeling of some special graphs”, International Journal of Scientific Research and Review, 7 (8) 2018.
[20]	S. N. Daoud and K. S. A, “Number of spanning trees of new join graphs” Algebra Letters, 2013.

Cite This Article

Plain Text BibTeX RIS

APA Style

Basma Mohamed, Mohamed Amin. (2023). Domination Number and Secure Resolving Sets in Cyclic Networks. Applied and Computational Mathematics, 12(2), 42-45. https://doi.org/10.11648/j.acm.20231202.12

Copy | Download

ACS Style

Basma Mohamed; Mohamed Amin. Domination Number and Secure Resolving Sets in Cyclic Networks. Appl. Comput. Math. 2023, 12(2), 42-45. doi: 10.11648/j.acm.20231202.12

Copy | Download

AMA Style

Basma Mohamed, Mohamed Amin. Domination Number and Secure Resolving Sets in Cyclic Networks. Appl Comput Math. 2023;12(2):42-45. doi: 10.11648/j.acm.20231202.12

Copy | Download

@article{10.11648/j.acm.20231202.12,
  author = {Basma Mohamed and Mohamed Amin},
  title = {Domination Number and Secure Resolving Sets in Cyclic Networks},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {2},
  pages = {42-45},
  doi = {10.11648/j.acm.20231202.12},
  url = {https://doi.org/10.11648/j.acm.20231202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231202.12},
  abstract = {Consider a robot that is navigating a graph-based space and is attempting to determine where it is right now. To determine how distant it is from each group of fixed landmarks, it can send a signal. We discuss the problem of determining the minimum number of landmarks necessary and their optimal placement to ensure that the robot can always locate itself. The number of landmarks is referred to as the graph's metric dimension, and the set of nodes on which they are distributed is known as the graph's metric basis. On the other hand, the metric dimension of a graph G is the minimum size of a set w of vertices that can identify each vertex pair of G by the shortest-path distance to a particular vertex in w. It is an NP-complete problem to determine the metric dimension for any network. The metric dimension is also used in a variety of applications, including geographic routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we investigate the exact value of the secure resolving set of some networks, such as trapezoid network, Z-(Pn) network, open ladder network, tortoise network and  network. We also determine the domination number of the networks, such as the twig network Tm, double fan network F2,n, bistar network Bn,n and linear kc4 – snake  network.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Domination Number and Secure Resolving Sets in Cyclic Networks
AU  - Basma Mohamed
AU  - Mohamed Amin
Y1  - 2023/05/29
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231202.12
DO  - 10.11648/j.acm.20231202.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 42
EP  - 45
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231202.12
AB  - Consider a robot that is navigating a graph-based space and is attempting to determine where it is right now. To determine how distant it is from each group of fixed landmarks, it can send a signal. We discuss the problem of determining the minimum number of landmarks necessary and their optimal placement to ensure that the robot can always locate itself. The number of landmarks is referred to as the graph's metric dimension, and the set of nodes on which they are distributed is known as the graph's metric basis. On the other hand, the metric dimension of a graph G is the minimum size of a set w of vertices that can identify each vertex pair of G by the shortest-path distance to a particular vertex in w. It is an NP-complete problem to determine the metric dimension for any network. The metric dimension is also used in a variety of applications, including geographic routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we investigate the exact value of the secure resolving set of some networks, such as trapezoid network, Z-(Pn) network, open ladder network, tortoise network and  network. We also determine the domination number of the networks, such as the twig network Tm, double fan network F2,n, bistar network Bn,n and linear kc4 – snake  network.
VL  - 12
IS  - 2
ER  -

Copy | Download

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

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Basma Mohamed, Mohamed Amin. (2023). Domination Number and Secure Resolving Sets in Cyclic Networks. Applied and Computational Mathematics, 12(2), 42-45. https://doi.org/10.11648/j.acm.20231202.12

Copy | Download

ACS Style

Basma Mohamed; Mohamed Amin. Domination Number and Secure Resolving Sets in Cyclic Networks. Appl. Comput. Math. 2023, 12(2), 42-45. doi: 10.11648/j.acm.20231202.12

Copy | Download

AMA Style

Basma Mohamed, Mohamed Amin. Domination Number and Secure Resolving Sets in Cyclic Networks. Appl Comput Math. 2023;12(2):42-45. doi: 10.11648/j.acm.20231202.12

Copy | Download

@article{10.11648/j.acm.20231202.12,
  author = {Basma Mohamed and Mohamed Amin},
  title = {Domination Number and Secure Resolving Sets in Cyclic Networks},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {2},
  pages = {42-45},
  doi = {10.11648/j.acm.20231202.12},
  url = {https://doi.org/10.11648/j.acm.20231202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231202.12},
  abstract = {Consider a robot that is navigating a graph-based space and is attempting to determine where it is right now. To determine how distant it is from each group of fixed landmarks, it can send a signal. We discuss the problem of determining the minimum number of landmarks necessary and their optimal placement to ensure that the robot can always locate itself. The number of landmarks is referred to as the graph's metric dimension, and the set of nodes on which they are distributed is known as the graph's metric basis. On the other hand, the metric dimension of a graph G is the minimum size of a set w of vertices that can identify each vertex pair of G by the shortest-path distance to a particular vertex in w. It is an NP-complete problem to determine the metric dimension for any network. The metric dimension is also used in a variety of applications, including geographic routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we investigate the exact value of the secure resolving set of some networks, such as trapezoid network, Z-(Pn) network, open ladder network, tortoise network and  network. We also determine the domination number of the networks, such as the twig network Tm, double fan network F2,n, bistar network Bn,n and linear kc4 – snake  network.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Domination Number and Secure Resolving Sets in Cyclic Networks
AU  - Basma Mohamed
AU  - Mohamed Amin
Y1  - 2023/05/29
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231202.12
DO  - 10.11648/j.acm.20231202.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 42
EP  - 45
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231202.12
AB  - Consider a robot that is navigating a graph-based space and is attempting to determine where it is right now. To determine how distant it is from each group of fixed landmarks, it can send a signal. We discuss the problem of determining the minimum number of landmarks necessary and their optimal placement to ensure that the robot can always locate itself. The number of landmarks is referred to as the graph's metric dimension, and the set of nodes on which they are distributed is known as the graph's metric basis. On the other hand, the metric dimension of a graph G is the minimum size of a set w of vertices that can identify each vertex pair of G by the shortest-path distance to a particular vertex in w. It is an NP-complete problem to determine the metric dimension for any network. The metric dimension is also used in a variety of applications, including geographic routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we investigate the exact value of the secure resolving set of some networks, such as trapezoid network, Z-(Pn) network, open ladder network, tortoise network and  network. We also determine the domination number of the networks, such as the twig network Tm, double fan network F2,n, bistar network Bn,n and linear kc4 – snake  network.
VL  - 12
IS  - 2
ER  -

Copy | Download