A new generalization of kernels in digraphs

Abstract

Let D be a digraph and k,l two positive integers. A subset N is a (k,l)-out-kernel of D if and only if N is a k-independent and a l-out-dominating set of D (that is Δ+(N)<k and ∀x∈V∖N,|ND+(x)∩N|≥l). A digraph such that every induced subdigraph has a (k,l)-out-kernel is called (k,l)-out-kernel perfect.A k-out-kernel is a (k,k)-out-kernel. Under this definition a kernel is a 1-out-kernel or a (1,1)-out-kernel.Since an (n−1)/2-regular digraph with an odd order does not have an (n−1)/2-out-kernel, the natural question is: which digraphs have a (k,l)-out-kernel or a k-out-kernel?In this paper we investigate the problem of the existence of a (k,l)-out-kernel and a k-out-kernel in digraphs, and generalize some classical results on kernels in digraphs.