On Eternal Domination of Generalized Js,m
An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the at...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/8882598 |
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| _version_ | 1832558414007369728 |
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| author | Ramy Shaheen Mohammad Assaad Ali Kassem |
| author_facet | Ramy Shaheen Mohammad Assaad Ali Kassem |
| author_sort | Ramy Shaheen |
| collection | DOAJ |
| description | An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex, and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m-eternal domination model. The size of the smallest m-eternal dominating set is called the m-eternal domination number and is denoted by γm∞G. In this paper, we find the domination number of Jahangir graph Js,m for s≡1,2 mod 3, and the m-eternal domination numbers of Js,m for s,m are arbitraries. |
| format | Article |
| id | doaj-art-cb636a1074454b68b38992fc6ff492b1 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-cb636a1074454b68b38992fc6ff492b12025-02-03T01:32:25ZengWileyJournal of Applied Mathematics1110-757X1687-00422021-01-01202110.1155/2021/88825988882598On Eternal Domination of Generalized Js,mRamy Shaheen0Mohammad Assaad1Ali Kassem2Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, SyriaDepartment of Mathematics, Faculty of Science, Tishreen University, Lattakia, SyriaDepartment of Mathematics, Faculty of Science, Tishreen University, Lattakia, SyriaAn eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex, and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m-eternal domination model. The size of the smallest m-eternal dominating set is called the m-eternal domination number and is denoted by γm∞G. In this paper, we find the domination number of Jahangir graph Js,m for s≡1,2 mod 3, and the m-eternal domination numbers of Js,m for s,m are arbitraries.http://dx.doi.org/10.1155/2021/8882598 |
| spellingShingle | Ramy Shaheen Mohammad Assaad Ali Kassem On Eternal Domination of Generalized Js,m Journal of Applied Mathematics |
| title | On Eternal Domination of Generalized Js,m |
| title_full | On Eternal Domination of Generalized Js,m |
| title_fullStr | On Eternal Domination of Generalized Js,m |
| title_full_unstemmed | On Eternal Domination of Generalized Js,m |
| title_short | On Eternal Domination of Generalized Js,m |
| title_sort | on eternal domination of generalized js m |
| url | http://dx.doi.org/10.1155/2021/8882598 |
| work_keys_str_mv | AT ramyshaheen oneternaldominationofgeneralizedjsm AT mohammadassaad oneternaldominationofgeneralizedjsm AT alikassem oneternaldominationofgeneralizedjsm |