One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set S⊆V is said to be dominating set of G if for every v є V-S there exists a vertex u є S such that uv є E. The dominance number of G, represented by γ(G), is the lowest cardinality of vertices among the dominating set of G. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph G and input K, γ(G) ≤ K. This is known as the dominating set issue. Consequently, it is thought that calculating γ(G) for each given graph G may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset S, then S⊆V is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of G is a set S of vertices in a graph G that is both an independent and a dominating set of G. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others.
Published in | Applied and Computational Mathematics (Volume 13, Issue 3) |
DOI | 10.11648/j.acm.20241303.11 |
Page(s) | 53-57 |
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. |
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Dominating Set, Domination Number, Triangular Snake Graph, Bistar Graph, Alternate Triangular Belt Graph, Line Graph, Crystal Planar Map, k-kite Chains Graph, Jewel Graph
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APA Style
Mohamed, B., Badawy, M. (2024). Some New Results on Domination and Independent Dominating Set of Some Graphs. Applied and Computational Mathematics, 13(3), 53-57. https://doi.org/10.11648/j.acm.20241303.11
ACS Style
Mohamed, B.; Badawy, M. Some New Results on Domination and Independent Dominating Set of Some Graphs. Appl. Comput. Math. 2024, 13(3), 53-57. doi: 10.11648/j.acm.20241303.11
AMA Style
Mohamed B, Badawy M. Some New Results on Domination and Independent Dominating Set of Some Graphs. Appl Comput Math. 2024;13(3):53-57. doi: 10.11648/j.acm.20241303.11
@article{10.11648/j.acm.20241303.11, author = {Basma Mohamed and Mohammed Badawy}, title = {Some New Results on Domination and Independent Dominating Set of Some Graphs }, journal = {Applied and Computational Mathematics}, volume = {13}, number = {3}, pages = {53-57}, doi = {10.11648/j.acm.20241303.11}, url = {https://doi.org/10.11648/j.acm.20241303.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241303.11}, abstract = {One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set S⊆V is said to be dominating set of G if for every v є V-S there exists a vertex u є S such that uv є E. The dominance number of G, represented by γ(G), is the lowest cardinality of vertices among the dominating set of G. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph G and input K, γ(G) ≤ K. This is known as the dominating set issue. Consequently, it is thought that calculating γ(G) for each given graph G may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset S, then S⊆V is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of G is a set S of vertices in a graph G that is both an independent and a dominating set of G. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others. }, year = {2024} }
TY - JOUR T1 - Some New Results on Domination and Independent Dominating Set of Some Graphs AU - Basma Mohamed AU - Mohammed Badawy Y1 - 2024/05/10 PY - 2024 N1 - https://doi.org/10.11648/j.acm.20241303.11 DO - 10.11648/j.acm.20241303.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 53 EP - 57 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20241303.11 AB - One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set S⊆V is said to be dominating set of G if for every v є V-S there exists a vertex u є S such that uv є E. The dominance number of G, represented by γ(G), is the lowest cardinality of vertices among the dominating set of G. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph G and input K, γ(G) ≤ K. This is known as the dominating set issue. Consequently, it is thought that calculating γ(G) for each given graph G may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset S, then S⊆V is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of G is a set S of vertices in a graph G that is both an independent and a dominating set of G. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others. VL - 13 IS - 3 ER -