Induced minors and well-quasi-ordering

A graph H is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor-free. Robin Thomas showed that K4-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without K4 and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 – 247, 1985]. We provide a dichotomy theorem for H-induced minor-free graphs and show that the class of H-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if H is an induced minor of the Gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489–502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasiordering, Journal of Graph Theory, 14(4):427–435, 1990].