Irreversible k -threshold conversion number of some graphs

• Published in: Arab journal of mathematical sciences Vol. 30; no. 1; pp. 43 - 56
• Main Authors: Shaheen, Ramy, Mahfud, Suhail, Kassem, Ali
• Format: Journal Article
• Language: English
• Published: Bingley Emerald Group Publishing Limited 23.01.2024; Emerald Publishing
• Subjects: Apexes; Bioterrorism; Graph conversion process; Graphs; Iterative methods; Jahangir graph; k-threshold conversion set; Social networks; Strong grid graph; Vertex sets
• ISSN: 1319-5166; 2588-9214
• DOI: 10.1108/AJMS-07-2021-0150

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.