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 graphi¾?H that Graph Isomorphism is polynomial-time solvable oni¾?H-induced-minor-free graphs or that it is isomorphism complete. Additionally, we classify those graphsi¾?H for which H-induced-minor-free graphs have bounded clique-width. Those two results complement similar dichotomies for graphs that exclude a fixed graph as an induced subgraph, minor or subgraph.

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