Irreversible k-threshold conversion number of some graphs
Purpose – This paper aims to study Irreversible conversion processes, which examine the spread of a one way change of state (from state 0 to state 1) through a specified society (the spread of disease through populations, the spread of opinion through social networks, etc.) where the conversion rule...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Emerald Publishing
2024-01-01
|
| Series: | Arab Journal of Mathematical Sciences |
| Subjects: | |
| Online Access: | https://www.emerald.com/insight/content/doi/10.1108/AJMS-07-2021-0150/full/pdf |
| _version_ | 1797348076534366208 |
|---|---|
| author | Ramy Shaheen Suhail Mahfud Ali Kassem |
| author_facet | Ramy Shaheen Suhail Mahfud Ali Kassem |
| author_sort | Ramy Shaheen |
| collection | DOAJ |
| description | Purpose – This paper aims to study Irreversible conversion processes, which examine the spread of a one way change of state (from state 0 to state 1) through a specified society (the spread of disease through populations, the spread of opinion through social networks, etc.) where the conversion rule is determined at the beginning of the study. These processes can be modeled into graph theoretical models where the vertex set V(G) represents the set of individuals on which the conversion is spreading. Design/methodology/approach – The irreversible k-threshold conversion process on a graph G=(V,E) is an iterative process which starts by choosing a set S_0?V, and for each step t (t = 1, 2,…,), S_t is obtained from S_(t−1) by adjoining all vertices that have at least k neighbors in S_(t−1). S_0 is called the seed set of the k-threshold conversion process and is called an irreversible k-threshold conversion set (IkCS) of G if S_t = V(G) for some t = 0. The minimum cardinality of all the IkCSs of G is referred to as the irreversible k-threshold conversion number of G and is denoted by C_k (G). Findings – In this paper the authors determine C_k (G) for generalized Jahangir graph J_(s,m) for 1 < k = m and s, m are arbitraries. The authors also determine C_k (G) for strong grids P_2? P_n when k = 4, 5. Finally, the authors determine C_2 (G) for P_n? P_n when n is arbitrary. Originality/value – This work is 100% original and has important use in real life problems like Anti-Bioterrorism. |
| first_indexed | 2024-03-08T11:58:17Z |
| format | Article |
| id | doaj.art-3b1ea0e23160474d9ed7c2c7a084c34e |
| institution | Directory Open Access Journal |
| issn | 1319-5166 2588-9214 |
| language | English |
| last_indexed | 2024-03-08T11:58:17Z |
| publishDate | 2024-01-01 |
| publisher | Emerald Publishing |
| record_format | Article |
| series | Arab Journal of Mathematical Sciences |
| spelling | doaj.art-3b1ea0e23160474d9ed7c2c7a084c34e2024-01-23T21:35:57ZengEmerald PublishingArab Journal of Mathematical Sciences1319-51662588-92142024-01-01301435610.1108/AJMS-07-2021-0150Irreversible k-threshold conversion number of some graphsRamy Shaheen0Suhail Mahfud1Ali Kassem2Department of Mathematics, Tishreen University, Lattakia, SyriaDepartment of Mathematics, Tishreen University, Lattakia, SyriaDepartment of Mathematics, Tishreen University, Lattakia, SyriaPurpose – This paper aims to study Irreversible conversion processes, which examine the spread of a one way change of state (from state 0 to state 1) through a specified society (the spread of disease through populations, the spread of opinion through social networks, etc.) where the conversion rule is determined at the beginning of the study. These processes can be modeled into graph theoretical models where the vertex set V(G) represents the set of individuals on which the conversion is spreading. Design/methodology/approach – The irreversible k-threshold conversion process on a graph G=(V,E) is an iterative process which starts by choosing a set S_0?V, and for each step t (t = 1, 2,…,), S_t is obtained from S_(t−1) by adjoining all vertices that have at least k neighbors in S_(t−1). S_0 is called the seed set of the k-threshold conversion process and is called an irreversible k-threshold conversion set (IkCS) of G if S_t = V(G) for some t = 0. The minimum cardinality of all the IkCSs of G is referred to as the irreversible k-threshold conversion number of G and is denoted by C_k (G). Findings – In this paper the authors determine C_k (G) for generalized Jahangir graph J_(s,m) for 1 < k = m and s, m are arbitraries. The authors also determine C_k (G) for strong grids P_2? P_n when k = 4, 5. Finally, the authors determine C_2 (G) for P_n? P_n when n is arbitrary. Originality/value – This work is 100% original and has important use in real life problems like Anti-Bioterrorism.https://www.emerald.com/insight/content/doi/10.1108/AJMS-07-2021-0150/full/pdfJahangir graphStrong grid graphGraph conversion processk-threshold conversion set |
| spellingShingle | Ramy Shaheen Suhail Mahfud Ali Kassem Irreversible k-threshold conversion number of some graphs Arab Journal of Mathematical Sciences Jahangir graph Strong grid graph Graph conversion process k-threshold conversion set |
| title | Irreversible k-threshold conversion number of some graphs |
| title_full | Irreversible k-threshold conversion number of some graphs |
| title_fullStr | Irreversible k-threshold conversion number of some graphs |
| title_full_unstemmed | Irreversible k-threshold conversion number of some graphs |
| title_short | Irreversible k-threshold conversion number of some graphs |
| title_sort | irreversible k threshold conversion number of some graphs |
| topic | Jahangir graph Strong grid graph Graph conversion process k-threshold conversion set |
| url | https://www.emerald.com/insight/content/doi/10.1108/AJMS-07-2021-0150/full/pdf |
| work_keys_str_mv | AT ramyshaheen irreversiblekthresholdconversionnumberofsomegraphs AT suhailmahfud irreversiblekthresholdconversionnumberofsomegraphs AT alikassem irreversiblekthresholdconversionnumberofsomegraphs |