Relation between number of kernels (and generalizations) of a digraph and its partial line digraphs

Abstract

Let D=(V,A) be a digraph, an arc subset A′⊆A and a surjective mapping ϕ:A→A′ such that the set of heads of A′ is V and ϕ|A′=Id and for every vertex j∈V, ϕ(ω−(j))⊂ω−(j)∩A′. The partial line digraph of D, LD, 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 δ−(D)≥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 every partial line digraph LD.