Irreversible k-Threshold Conversion Number of Circulant Graphs
An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k-threshold conversion process on a graph G=V,E...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2022-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/1250951 |
| _version_ | 1798033262022492160 |
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| author | Ramy Shaheen Suhail Mahfud Ali Kassem |
| author_facet | Ramy Shaheen Suhail Mahfud Ali Kassem |
| author_sort | Ramy Shaheen |
| collection | DOAJ |
| description | An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k-threshold conversion process on a graph G=V,E is an iterative process which begins by choosing a set S0⊆V, and for each step tt=1,2,⋯,,St is obtained from St−1 by adjoining all vertices that have at least k neighbors in St−1. S0 is called the seed set of the k-threshold conversion process, and if St=VG for some t≥0, then S0 is an irreversible k-threshold conversion set (IkCS) of G. The k-threshold conversion number of G (denoted by (CkG) is the minimum cardinality of all the IkCSs of G. In this paper, we determine C2G for the circulant graph Cn1,r when r is arbitrary; we also find C3Cn1,r when r=2,3. We also introduce an upper bound for C3Cn1,4. Finally, we suggest an upper bound for C3Cn1,r if n≥2r+1 and n≡0mod 2r+1. |
| first_indexed | 2024-04-11T20:26:26Z |
| format | Article |
| id | doaj.art-d01e0584b6c24bdda28e00fd285d1385 |
| institution | Directory Open Access Journal |
| issn | 1687-0042 |
| language | English |
| last_indexed | 2024-04-11T20:26:26Z |
| publishDate | 2022-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj.art-d01e0584b6c24bdda28e00fd285d13852022-12-22T04:04:38ZengHindawi LimitedJournal of Applied Mathematics1687-00422022-01-01202210.1155/2022/1250951Irreversible k-Threshold Conversion Number of Circulant GraphsRamy Shaheen0Suhail Mahfud1Ali Kassem2Department of MathematicsDepartment of MathematicsDepartment of MathematicsAn irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k-threshold conversion process on a graph G=V,E is an iterative process which begins by choosing a set S0⊆V, and for each step tt=1,2,⋯,,St is obtained from St−1 by adjoining all vertices that have at least k neighbors in St−1. S0 is called the seed set of the k-threshold conversion process, and if St=VG for some t≥0, then S0 is an irreversible k-threshold conversion set (IkCS) of G. The k-threshold conversion number of G (denoted by (CkG) is the minimum cardinality of all the IkCSs of G. In this paper, we determine C2G for the circulant graph Cn1,r when r is arbitrary; we also find C3Cn1,r when r=2,3. We also introduce an upper bound for C3Cn1,4. Finally, we suggest an upper bound for C3Cn1,r if n≥2r+1 and n≡0mod 2r+1.http://dx.doi.org/10.1155/2022/1250951 |
| spellingShingle | Ramy Shaheen Suhail Mahfud Ali Kassem Irreversible k-Threshold Conversion Number of Circulant Graphs Journal of Applied Mathematics |
| title | Irreversible k-Threshold Conversion Number of Circulant Graphs |
| title_full | Irreversible k-Threshold Conversion Number of Circulant Graphs |
| title_fullStr | Irreversible k-Threshold Conversion Number of Circulant Graphs |
| title_full_unstemmed | Irreversible k-Threshold Conversion Number of Circulant Graphs |
| title_short | Irreversible k-Threshold Conversion Number of Circulant Graphs |
| title_sort | irreversible k threshold conversion number of circulant graphs |
| url | http://dx.doi.org/10.1155/2022/1250951 |
| work_keys_str_mv | AT ramyshaheen irreversiblekthresholdconversionnumberofcirculantgraphs AT suhailmahfud irreversiblekthresholdconversionnumberofcirculantgraphs AT alikassem irreversiblekthresholdconversionnumberofcirculantgraphs |