Given an $H$-induced-minor-free graph $G$ with $n$ vertices, we construct a balanced separator of $G$ consisting of $\tilde{O}(|V(H)|\sqrt{n})$ star subgraphs. Our result is an induced-minor-free analog of the famous separator theorem for $H$-minor-free graphs by Alon, Seymour, and Thomas [J. Amer. Math. Soc., 1990]. A key idea in our construction is a new decomposition of $H$-induced-minor-free graphs, analogous to the cop decomposition for $H$-minor-free graphs by Andreae [J. Combin. Theory Ser. B, 1986]. We also discuss several algorithmic applications of our result.

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