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Revisiting classical results on kernels in digraphs

Hélène Langlois, Frédéric Meunier

TL;DR

This work revisits three classical sufficient conditions for the existence of kernels in digraphs—red-blue arc structures, chords in odd holes, and clique-acyclic orientations of anti-holes—and provides broad generalizations that preserve polynomial-time solvability. It introduces a generalized red-blue framework, utilizes poset-based antichain techniques and semi-kernel arguments to construct kernels, and extends known results beyond traditional transitivity constraints. The paper also clarifies the kernel-solvability landscape for anti-holes, demonstrating that odd anti-holes with $n\ge 9$ are simple kernel-solvable but not kernel-solvable, thereby refining connections to perfect graphs and highlighting open algorithmic questions. Collectively, these contributions deepen understanding of kernel existence criteria and offer new, tractable avenues for identifying kernels in broader digraph classes.

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-Sánchez--Neumann-Lara theorem.

Revisiting classical results on kernels in digraphs

TL;DR

This work revisits three classical sufficient conditions for the existence of kernels in digraphs—red-blue arc structures, chords in odd holes, and clique-acyclic orientations of anti-holes—and provides broad generalizations that preserve polynomial-time solvability. It introduces a generalized red-blue framework, utilizes poset-based antichain techniques and semi-kernel arguments to construct kernels, and extends known results beyond traditional transitivity constraints. The paper also clarifies the kernel-solvability landscape for anti-holes, demonstrating that odd anti-holes with are simple kernel-solvable but not kernel-solvable, thereby refining connections to perfect graphs and highlighting open algorithmic questions. Collectively, these contributions deepen understanding of kernel existence criteria and offer new, tractable avenues for identifying kernels in broader digraph classes.

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-Sánchez--Neumann-Lara theorem.

Paper Structure

This paper contains 17 sections, 12 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Digraphs with red and blue arcs
  3. Chords in odd holes
  4. Clique-acyclic orientations of anti-holes
  5. Some terminology and notation
  6. "Red--blue" arcs
  7. Proof of Theorem \ref{['thm:br']}
  8. The poset of antichains
  9. Two lemmas
  10. The proof
  11. Discussion
  12. Odd holes
  13. Proof of Theorem \ref{['thm:cordes_nous']}
  14. Semi-kernels
  15. The proof
  16. ...and 2 more sections

Key Result

Theorem 1

Let $D$ be a digraph whose arcs are colored with two colors. Then there is a subset $S$ of vertices such that no two vertices in $S$ are connected by a monochromatic directed path and such that from every vertex there is a monochromatic directed path ending in $S$.

Figures (5)

  • Figure 1: The condition in Theorem \ref{['thm:br']}
  • Figure 2: An odd directed cycle having two odd chords that are neither crossing nor nested
  • Figure 3: Odd directed cycles having two crossing chords, one short and the other being odd
  • Figure 4: The induced structures forbidden by condition \ref{['cond4br']} in Proposition \ref{['prop:4br']}
  • Figure 5: A simple clique-acyclic orientation of $\overline{C}_{7}$ with no kernel.

Theorems & Definitions (22)

  • Theorem : Sands, Sauer, and Woodrow sands1982monochromatic
  • Theorem 1.1
  • Theorem : Galeana-Sánchez and Neumann-Lara galeana1984kernels
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 12 more