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 kc_{4} – 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), 2023. Published by Science Publishing Group |
Domination Number, Secure Resolving Set, Twig Graph and Linear kc4 - Snake Graph
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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
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
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
@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} }
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 -