(k,l)-kernels in line digraphs

Let D=(V,A) be a digraph with minimum indegree at least one and girth at least k, where k⩾2 is an integer. In this paper , the following results were proved. A digraph D has a (k,l)-kernel if and only if its line digraph L(D) does, and the number of (k,l)-kernels in D is equal to the number of (k,l)-kernels in L(D), where 1⩽l<k. As a consequence, previous results about k-kernels in the line digraphs were gained.