Before revision:  At the heart of our algorithm are a combination of two kinds of tree-decompositions and a novel structural theorem that generalizes a theorem in Mukhopadhyay-Nanongkai [STOC'20] and , in turn, helps simplify their algorithms .
After revision:  At the heart of our algorithm are a combination of two kinds of tree-decompositions and a novel structural theorem that generalizes a theorem in Mukhopadhyay-Nanongkai [STOC'20] and can be of independent interest .