Kernels and partial line digraphs

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 L D 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.