Well-quasi-ordering H-contraction-free graphs

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 of the previous dichotomy theorems of Damsaschke (1990) on the induced subgraph relation, Ding (1992) on the subgraph relation, and Błasiok et al. (2015) on the induced minor relation.