Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/3344 2022 2022 Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria Ali Ali Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria Ramy Shaheen Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria Suhail Mahfud An irreversible k-threshold conversion process on a graph 𝐺=(𝑉,𝐸) is a dynamic, iterative process which begins by choosing a set 𝑆0⊆𝑉. For each step 𝑡(𝑡=1,2,…,), 𝑆𝑡 is obtained from 𝑆𝑡−1 by adjoining all vertices that have at least k neighbors in 𝑆𝑡−1. We call 𝑆0 the seed set of the k-threshold conversion process and if 𝑆𝑡=𝑉(𝐺) for some 𝑡≥0, then 𝑆0 is called an irreversible k-threshold conversion set (IkCS) of 𝐺. The k-threshold conversion number of 𝐺 (denoted by (𝐶𝑘(𝐺)) is the minimum cardinality of all the IkCSs of 𝐺. In this paper, we study Irreversible k-threshold conversion processes on strong grids 𝑃𝑚⊠𝑃𝑛. We determine 𝐶𝑘(𝑃3⊠𝑃𝑛) for 𝑘=5,6,7 and 𝐶𝑘(𝑃4⊠𝑃𝑛) for 𝑘=6,7. We also present upper bounds for 𝐶4(𝑃3⊠𝑃𝑛), 𝐶4(𝑃4⊠𝑃𝑛),𝐶5(𝑃3⊠𝑃𝑛), then we determine 𝐶8(𝑃𝑚⊠𝑃𝑛) for arbitrary 𝑚,𝑛. 2024 2024 60 72 10.54216/GJMSA.0110208 https://www.americaspg.com/articleinfo/33/show/3344