Induced minors and well-quasi-ordering

TL;DR

This paper characterizes exactly which graphs have classes of graphs that are well-quasi-ordered by induced minors, providing a clear dichotomy based on induced minor containment.

Contribution

It establishes a dichotomy theorem for $H$-induced minor-free graphs, identifying the specific graphs $H$ for which the class is well-quasi-ordered, and introduces two new decomposition theorems.

Findings

Class of $H$-induced minor-free graphs is well-quasi-ordered iff $H$ is an induced minor of the gem or a specific 4-clique extension.

Provides a complete characterization for well-quasi-ordering in the context of induced minors.

Introduces two decomposition theorems of independent interest.

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 that $K_4$-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without $K_4$ 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-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990].