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

TL;DR

This paper investigates how certain kernel and Grundy function properties in digraphs are preserved or related when constructing their partial line digraphs, providing bounds and equalities for these combinatorial structures.

Contribution

It introduces the concept of $(k,l)$-Grundy functions and establishes relationships between the number of kernels and Grundy functions in a digraph and its partial line digraph.

Findings

Number of $(k,l)$-kernels in $D$ is at most that in $ ilde{D}$.

Equal number of semikernels in $D$ and $ ilde{D}$.

Number of $(k,l)$-Grundy functions is preserved 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 $\mathcal{L} D$.