Semikernels modulo \(F\) and kernels in digraphs (Q1978146)

Let \(D=(V,A)\) be a finite digraph and let \(V\) and \(A\) denote the set of all vertices and arcs of \(D,\) respectively. A set \(S\) of vertices is defined to be a semikernel of \(D\) modulo \(F\) if \(S\subseteq V,\) \(F\subseteq A,\) \(S\) is independent in \(D\) and for every \(z\in V-S\) for which there is an \(Sz\)-arc belonging to \(D-F,\) there exists a \(zS\)-arc in \(A.\) A new sufficient condition for a digraph to have a kernel is obtained and a result of \textit{B. Sands, N. Sauer} and \textit{R. Woodrow} on monochromatic paths [J. Comb. Theory, Ser. B 33, 271-275 (1982; Zbl 0488.05036)] is generalized.