On the Irreversible k-Threshold Conversion Number for Some Graph Products and Neutrosophic Graphs

Ahmad A. Abubaker^1.*, Raed Hatamleh², Khaled Matarneh¹, Abdallah Al-Husban^{3, 4}

¹Faculty of Computer Studies, Arab Open University, Saudi Arabia
²Department of Mathematics, Faculty of Science and Information Technology, Jadara University, P.O. Box 733, Irbid 21110, Jordan

³Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan
⁴Jadara Research Center, Jadara University, Irbid 21110, Jordan

Emails: a.abubaker@arabou.edu.sa; raed@jadara.edu.jo; k.matarneh@arabou.edu.sa; dralhosban@inu.edu.jo

Abstract

An irreversible k-threshold conversion process on a graph  is an iterative process that studies the spread of a one way change (from state 0 to 1) on . The process begins by choosing a set . For each step    is obtained from  by adjoining all vertices that have at least k neighbors in . We call  the seed set of the k-threshold conversion process and if  for some , then  is called an irreversible k-threshold conversion set (IkCS) of . The k-threshold conversion number of  (denoted by ( ) is the minimum cardinality of all the IkCSs of . In this paper, we study IkCSs of toroidal grids and the tensor product of two paths. We determine and we present upper and lower bounds for  for . We also determine  and present an upper bound for  when . Then we determine  for  and arbitrary . Finally, we determine  for arbitrary . . Also, we study the same concepts over some neutrosophic graphs with suggestions for future neutrosophic and fuzzy generalizations.

Keywords: Toroidal grid; Tensor product; Graph conversion process; k-threshold conversion set; Neutrosophic graph; Neutrosophic graph product