Induced Minor Free Graphs: Isomorphism and Clique-Width

Given two graphs G and H, we say that G contains H as an induced minor if a graph isomorphic to H can be obtained from G by a sequence of vertex deletions and edge contractions. We study the complexity of Graph Isomorphism on graphs that exclude a fixed graph as an induced minor. More precisely, we determine for every graph H that Graph Isomorphism is polynomial-time solvable on H-induced-minor-free graphs or that it is GI-complete. Additionally, we classify those graphs H for which H-induced-minor-free graphs have bounded clique-width. These two results complement similar dichotomies for graphs that exclude a fixed graph as an induced subgraph, minor, or subgraph.