On exponential domination of the consecutive circulant graph

Abstract: For a graph $G,$ we consider $D \subset V(G)$ to be a porous exponential dominating set if $1\le \sum_{d \in D}$ $\left( \frac{1}{2} \right)^{\text{dist}(d,v) -1}$ for every $v \in V(G),$ where dist$(d,v)$ denotes the length of the smallest $dv$ path. Similarly, $D \subset V(G)$ is a non-porous exponential dominating set is $1\le \sum_{d \in D} \left( \frac{1}{2} \right)^{\overline{\text{dist}}(d,v) -1}$ for every $v \in V(G),$ where $\overline{\text{dist}}(d,v)$ represents the length of the shortest $dv$ path with no internal vertices in $D.$ The porous and non-porous exponential dominating number of $G,$ denoted $\gamma_e^*(G)$ and $\gamma_e(G),$ are the minimum cardinality of a porous and non-porous exponential dominating set, respectively. The consecutive circulant graph, $C_{n, [\ell]},$ is the set of $n$ vertices such that vertex $v$ is adjacent to $v \pm i \mod n$ for each $i \in [\ell].$ In this paper we show $\gamma_e(C_{n, [\ell]}) = \gamma_e^*(C_{n, [\ell]}) = \left\lceil \tfrac{n}{3\ell +1} \right\rceil.$