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 Digraphs - theory, algorithms and applications, Second Edition, Springer-Verlag, 2009, by J. Bang-Jensen and G. Gutin.
J. Bang-Jensen and G. Gutin (2009). Digraphs - theory, algorithms and applications. Second Edition, Springer-Verlag. J. Bang-Jensen, Y. Guo, G. Gutin and L. Volkmann (1997). A classification of locally semi-complete digraphs. Discrete Math.. 167/168, 101-114 M. R. Garey and D. S. Johnson (1979). Computers and intractability. A Series of Books in the Mathematical Sciences. W. H. Freeman and Co., San Francisco, Calif.. J. von Neumann and O. Morgenstern (1944). Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ.
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