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I recommend reading Sections 7 and 14 in the excellent book by Cygan, Fomin, Kowalik, Lokshtanov, Marx, Pilipczuk, Pilipczuk, and Saurabh. In short, Gu and Tamaki give a quadratic time algorithm which finds a branch-decomposition of a planar graph of width at most $3\sqrt{n}$. Then Robertson and Seymour in (5.1) give a tree-decomposition of width less than ...

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