Irreversible k‐Threshold Conversion Number of Circulant Graphs

Ramy Shaheen, Suhail Mahfud and Ali Kassem

Journal of Applied Mathematics, 2022, vol. 2022, issue 1

Abstract: An irreversible conversion process is a dynamic process on a graph where a one‐way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k‐threshold conversion process on a graph G = (V, E) is an iterative process which begins by choosing a set S0⊆V, and 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. S0 is called the seed set of the k‐threshold conversion process, and if St = V(G) for some t ≥ 0, then S0 is 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 determine C2(G) for the circulant graph Cn({1, r}) when r is arbitrary; we also find C3(Cn({1, r})) when r = 2, 3. We also introduce an upper bound for C3(Cn({1, 4})). Finally, we suggest an upper bound for C3(Cn({1, r})) if n ≥ 2(r + 1) and n ≡ 0(mod 2(r + 1)).

Date: 2022
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https://doi.org/10.1155/2022/1250951

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2022:y:2022:i:1:n:1250951

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