$$\varPi $$ -Kernels in Digraphs.

Let $$D=(V(D), A(D))$$ be a digraph, $$DP(D)$$ be the set of directed paths of $$D$$ and let $$\varPi $$ be a subset of $$DP(D)$$ . A subset $$S\subseteq V(D)$$ will be called $$\varPi $$ -independent if for any pair $$\{x, y\} \subseteq S$$ , there is no $$xy$$ -path nor $$yx$$ -path in $$\varPi $$ ; and will be called $$\varPi $$ -absorbing if for every $$x\in V(D)\setminus S$$ there is $$y\in S$$ such that there is an $$xy$$ -path in $$\varPi $$ . A set $$S\subseteq V(D)$$ will be called a $$\varPi $$ -kernel if $$S$$ is $$\varPi $$ -independent and $$\varPi $$ -absorbing. This concept generalize several 'kernel notions' like kernel or kernel by monochromatic paths, among others. In this paper we present some sufficient conditions for the existence of $$\varPi $$ -kernels. [ABSTRACT FROM AUTHOR]