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.