About (k,l)-kernels, semikernels and Grundy functions in partial line digraphs

Abstract

Let D be a digraph of minimum in-degree at least 1. We prove that for any two natural numbers k, l such that 1 = l = k, the number of (k, l)-kernels of D is less than or equal to the number of (k, l)-kernels of any partial linedigraph LD. Moreover, if l < k and the girth of D is at least l+1, then these two numbers are equal. We also prove that the number of semikernels of D is equal to the number of semikernels of LD. Furthermore, we introduce the concept of (k, l)-Grundy function as a generalization of the concept of Grundy function and we prove that the number of (k, l)-Grundy functions of D is equal to the number of (k, l)-Grundy functions of any partial line digraph LD.