Kernels and partial line digraphs

Abstract

Let D = (V,A) be a digraph with minimum in-degree at least 1 and girth at least l+1, where l ≥ 1. In this work, the following result is proved: a digraph D has a (k,l)-kernel if and only if its partial line digraph LD does, where 1 ≤ l < k. As a consequence, the h-iterated line digraph $L^h$(D) is shown to have a kernel if and only if D has a kernel.