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 $G=\left(V,E\right)$ is an iterative process which begins by choosing a set $S_0\subseteq V$ , and for each step $t\left(t=1,2,\cdots ,\right),$ $S_t$ 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 $S_t=V\left(G\right)$ for some $t\ge 0$ , then $S_0$ is an irreversible $k$ -threshold conversion set (IkCS) of $G$ . The $k$ -threshold conversion number of $G$ (denoted by ( $C_k\left(G\right)$ ) is the minimum cardinality of all the IkCSs of $G.$ In this paper, we determine $C_2\left(G\right)$ for the circulant graph $C_n\left(\left\{1,\text{r}\right\}\right)$ when $r$ is arbitrary; we also find $C_3\left(C_n\left(\left\{1,\text{r}\right\}\right)\right)$ when $r=2,3$ . We also introduce an upper bound for $C_3\left(C_n\left(\left\{1,4\right\}\right)\right)$ . Finally, we suggest an upper bound for $C_3\left(C_n\left(\left\{1,\text{r}\right\}\right)\right)$ if $n\ge 2\left(r+1\right)$ and $n\equiv 0\left(\mathrm{mod}2\left(r+1\right)\right)$ .