Kernels in circulant digraphs

A kernel $J$ of a digraph $D$ is an independent set of vertices of $D$ such that for every vertex $w,in,V(D),setminus,J$ there exists an arc from $w$ to a vertex in $J.$ In this paper, among other results, a characterization of $2$-regular circulant digraph having a kernel is obtained. This characterization is a partial solution to the following problem: Characterize circulant digraphs which have kernels; it appeared in the book {it Digraphs - theory, algorithms and applications}, Second Edition, Springer-Verlag, 2009, by J. Bang-Jensen and G. Gutin.