On exponential domination of the consecutive circulant graph

For a graph $G,$ we consider $D\subset V(G)$ to be a porous exponential dominating set if $1\le {\sum }_{d\in D}$ $\left( \frac{1}{2} \right)^{$\text{dist}$(d,v) -1}$forevery$v \in V(G),$wheredist$(d,v)$denotesthelengthofthesmallest$dv$path\mathrm{.}Similarly,$D \subset V(G)$ is a non-porous exponential dominating set is $1\le \sum_{d \in D} \left( \frac{1}{2} \right)^{\overline{$\text{dist}$}(d,v) -1}$forevery$v \in V(G),$where$\overline{$\text{dist}$}(d,v)$representsthelengthoftheshortest$dv$pathwithnointernalverticesin$D.$ The porous and non-porous exponential dominating number of $G,$ denoted ${\gamma }_e^{\ast }(G)$ and ${\gamma }_e(G),$ are the minimum cardinality of a porous and non-porous exponential dominating set, respectively. The consecutive circulant graph, $C_{n,[\mathrm{\ell }]},$ is the set of $n$ vertices such that vertex $v$ is adjacent to $v\pm i\mathrm{m}\mathrm{o}\mathrm{d}n$ for each $i \in [\ell].$Inthispaperweshow$\gamma_e(C_{n, [\ell]}) = \gamma_e^*(C_{n, [\ell]}) = \left\lceil \tfrac{n}{3\ell +1} \right\rceil.$