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 $A'\subseteq A$ and an exhaustive mapping $\phi: A\to A'$ such that $(i)$ the set of heads of $A'$ is $H(A')=V$; $(ii)$ the map fixes the elements of $A'$, that is, $\phi|A'=Id$, and for every vertex $j\in V$, $\phi(\omega^-(j))\subset \omega^-(j)\cap A'$. Then, {\it the partial line digraph} of $D$, denoted by $\mathcal{L}_{(A',\phi)}D $ (for short $\mathcal{L}D$ if the pair $(A', \phi)$ is clear from the context), is the digraph with vertex set $V (\mathcal{L}D)=A'$ and set of arcs $A(\mathcal{L}D) = \{(ij, \phi(j,k)) : (j,k)\in A\}.$ In this paper we prove the following results: Let $k,l$ be two natural numbers such that $1\le l \le k$, 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 $\mathcal{L} D$. 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 $\mathcal{L} D$. 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 $\mathcal{L} D$.