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 ${\gamma }_e^{\ast }(G)$ and ${\gamma }_e(G),$ are the minimum cardinality of a porous and non-porous exponential dominating set, respectively. The consecutive circulant graph, $C_{n,[\ell ]},$ is the set of $n$ vertices such that vertex $v$ is adjacent to $v\pm i\mathrm{mod}n$ for each $i\in [\ell ].$ In this paper we show ${\gamma }_e(C_{n,[\ell ]})={\gamma }_e^{\ast }(C_{n,[\ell ]})=\lceil \frac{n}{3\ell +1}\rceil .$