The class of graphs with no induced subdivision of H has bounded clique-width if and only if H is an induced subgraph of P4, paw, or diamond.
Dabrowski and Daniël Paulusma
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Clique-width and induced topological minors
The class of graphs with no induced subdivision of H has bounded clique-width if and only if H is an induced subgraph of P4, paw, or diamond.