Well-quasi-ordering H-contraction-free graphs

TL;DR

This paper characterizes exactly which graphs H lead to classes of H-contraction-free graphs being well-quasi-ordered under contraction, extending previous results on subgraph and minor relations.

Contribution

It provides a complete characterization of graphs H for which H-contraction-free graphs form a well-quasi-order, advancing understanding of graph contraction relations.

Findings

Identifies graphs H that induce well-quasi-ordered classes

Extends previous dichotomy theorems to contraction relation

Completes the classification for H-contraction-free graph classes

Abstract

A well-quasi-order is an order which contains no infinite decreasing sequence and no infinite collection of incomparable elements. In this paper, we consider graph classes defined by excluding one graph as contraction. More precisely, we give a complete characterization of graphs H such that the class of H-contraction-free graphs is well-quasi-ordered by the contraction relation. This result is the contraction analogue on the previous dichotomy theorems of Damsaschke [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427-435, 1990] on the induced subgraph relation, Ding [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489-502, 1992] on the subgraph relation, and B{\l}asiok et al. [Induced minors and well-quasi-ordering, ArXiv e-prints, 1510.07135, 2015] on the induced minor relation.