Revisiting classical results on kernels in digraphs

TL;DR

This paper revisits and generalizes classical theorems on the existence of kernels in directed graphs, providing deeper theoretical insights into conditions that guarantee kernel existence.

Contribution

It extends key classical results like the Sands--Sauer--Woodrow and Galeana-Sánchez--Neumann-Lara theorems, broadening their applicability in digraph theory.

Findings

Generalized classical kernel existence theorems

Provided new sufficient conditions for kernels in digraphs

Enhanced understanding of kernel properties in directed graphs

Abstract

In a digraph, a kernel is a subset of vertices that is both independent and absorbing. Kernels have important applications in combinatorics and outside. Kernels do not always exist and finding sufficient conditions ensuring their existence is a key theoretical challenge. In this work, we revisit and generalize a few classical results of this sort, especially the Sands--Sauer--Woodrow theorem and the Galeana-S\'anchez--Neumann-Lara theorem.