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

Abstract

Let D = (V, A) be a digraph and consider an arc subset A ′ ⊆ A and an exhaustive mapping φ : A → A ′ such that (i) the set of heads of A ′ is H(A ′ ) = V ;(ii) the map fixes the elements of A ′ , that is, φ|A ′ = Id, and for every vertex j ∈ V ,Then, the partial line digraph of D, denoted by L (A ′ ,φ) D (for short LD if the pair (A ′ , φ) is clear from the context), is the digraph with vertex set V (LD) = A ′ and set of arcs A(LD) = {(ij, φ(j, k)) : (j, k) ∈ A}. In this paper we prove the following results: Let k, l be two natural numbers such that 1 ≤ l ≤ k, and D a digraph with minimum in-degree at least 1. Then the number of (k, l)-kernels of D is less than or equal to the number of (k, l)-kernels of LD. Moreover, if l < k and the girth of D is at least l + 1, then these two numbers are equal.The number of semikernels of D is equal to the number of semikernels of LD. Also 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.