Product structure of graphs with an excluded minor - Illingworth, F; Scott, A; Wood, D (American Mathematical Society, 2024)
This paper shows that Kt-minor-free (and Ks,t-minor-free) graphs G are subgraphs of products of a tree-like graph H (of bounded treewidth) and a complete graph Km. Our results include optimal bounds on the treewidth of H and optimal bounds (to within a constant factor) on m in terms of the number of vertices of G and the treewidth of G. These results follow from a more general theorem whose corollaries include a strengthening of the celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc. 1990] and the Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM 2020].