Irreversible k -Threshold Conversion Number of Circulant Graphs

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 = ð V , E Þ is an iterative process which begins by choosing a set S 0 ⊆ V , and for each step t ð t = 1, 2, ⋯ , Þ , S t is obtained from S t − 1 by adjoining all vertices that have at least k neighbors in S t − 1 . S 0 is called the seed set of the k -threshold conversion process, and if S t = V ð G Þ for some t ≥ 0 , then S 0 is an irreversible k -threshold conversion set (IkCS) of G . The k -threshold conversion number of G (denoted by ( C k ð G Þ ) is the minimum cardinality of all the IkCSs of G : In this paper, we determine C 2 ð G Þ for the circulant graph C n ðf 1, r gÞ when r is arbitrary; we also fi nd C 3 ð C n ðf 1, r gÞÞ when r = 2, 3 . We also introduce an upper bound for C 3 ð C n ðf 1, 4 gÞÞ . Finally, we suggest an upper bound for C 3 ð C n ðf 1, r gÞÞ if n ≥ 2 ð r + 1 Þ and n ≡ 0 ð mod2 ð r + 1 ÞÞ .