$\varPi$-Kernels in Digraphs

Abstract

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.