Tournament Minors

TL;DR

This paper proves that the class of all tournaments is well-quasi-ordered under the minor relation defined by contracting strongly-connected subdigraphs, extending the understanding of structural orderings in directed graphs.

Contribution

It establishes that all tournaments are well-quasi-ordered under a new minor relation involving contractions of strongly-connected subdigraphs, a novel structural result.

Findings

The class of all tournaments is a well-quasi-order under the defined minor relation.

The minor relation involves contracting strongly-connected subdigraphs to vertices.

This result extends the theory of graph minors to directed graphs, specifically tournaments.

Abstract

We say a digraph $G$ is a {\em minor} of a digraph $H$ if $G$ can be obtained from a subdigraph of $H$ by repeatedly contracting a strongly-connected subdigraph to a vertex. Here, we show the class of all tournaments is a well-quasi-order under minor containment.