Volume 25 , Issue 2 , PP: 183-196, 2025 | Cite this article as | XML | Html | PDF | Full Length Article
Ahmad A. Abubaker 1 * , Raed Hatamleh 2 , Khaled Matarneh 3 , Abdallah Al-Husban 4
Doi: https://doi.org/10.54216/IJNS.250216
An irreversible k-threshold conversion process on a graph G=(V,E) is an iterative process that studies the spread of a one way change (from state 0 to 1) on V(G). The process begins by choosing a set S_0⊆V. 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). We call S_0 the seed set of the k-threshold conversion process and if S_t=V(G) for some t≥0, then S_0 is called an irreversible k-threshold conversion set (IkCS) of G. The k-threshold conversion number of G (denoted by (c_k (G)) is the minimum cardinality of all the IkCSs of G. In this paper, we study IkCSs of toroidal grids and the tensor product of two paths. We determine c_2 (C_3×C_n ) and we present upper and lower bounds for c_2 (C_m×C_n) for m,n≥3. We also determine c_2 (P_2×P_n ),c_2 (P_3×P_n ) and present an upper bound for c_2 (P_m×P_n) when m,n>3. Then we determine c_3 (P_m×P_n) for m=2,3,4 and arbitrary n. Finally, we determine c_4 (P_m×P_n) for arbitrary m,n. . Also, we study the same concepts over some neutrosophic graphs with suggestions for future neutrosophic and fuzzy generalizations.
Toroidal grid , Tensor product , Graph conversion process , k-threshold conversion set , Neutrosophic graph , Neutrosophic graph product
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