Induced minors and well-quasi-ordering

Abstract : 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 in [Graphs without K 4 and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 – 247, 1985] that K 4-induced minor-free graphs are well-quasi ordered by induced minors. 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. Similar dichotomy results were previously given by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489–502, 1992] for subgraphs and Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427–435, 1990] for induced subgraphs.
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Jarosław Błasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck. Induced minors and well-quasi-ordering. EuroComb: European Conference on Combinatorics, Graph Theory and Applications, Aug 2015, Bergen, Norway. pp.197-201, ⟨10.1016/j.endm.2015.06.029⟩. ⟨lirmm-01349277⟩

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