Galoitica: Journal of Mathematical Structures and Applications
Journal DOI
2834-5568ISSN (Online)
Volume 11 , Issue 2 , PP: 60-72, 2024 | Cite this article as | XML | Html | PDF | Full Length Article
Ali Kassem 1 * , Ramy Shaheen 2 , Suhail Mahfud 3
Doi: https://doi.org/10.54216/GJMSA.0110208
An irreversible k-threshold conversion process on a graph πΊ=(π,πΈ) is a dynamic, iterative process which begins by choosing a set π0⊆π. For each step π‘(π‘=1,2,…,), ππ‘ is obtained from ππ‘−1 by adjoining all vertices that have at least k neighbors in ππ‘−1. We call π0 the seed set of the k-threshold conversion process and if ππ‘=π(πΊ) for some π‘≥0, then π0 is called an irreversible k-threshold conversion set (IkCS) of πΊ. The k-threshold conversion number of πΊ (denoted by (πΆπ(πΊ)) is the minimum cardinality of all the IkCSs of πΊ. In this paper, we study Irreversible k-threshold conversion processes on strong grids ππβ ππ. We determine πΆπ(π3β ππ) for π=5,6,7 and πΆπ(π4β ππ) for π=6,7. We also present upper bounds for πΆ4(π3β ππ), πΆ4(π4β ππ),πΆ5(π3β ππ), then we determine πΆ8(ππβ ππ) for arbitrary π,π.
Strong grid , graph conversion process , k-threshold conversion set
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