  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].
