Product structure of graphs with an excluded minor

Abstract

This paper shows that K t K_t -minor-free (and K s , t K_{s, t} -minor-free) graphs G G are subgraphs of products of a tree-like graph H H (of bounded treewidth) and a complete graph K m K_m . Our results include optimal bounds on the treewidth of H H and optimal bounds (to within a constant factor) on m m in terms of the number of vertices of G G and the treewidth of G 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. 3 (1990), 801–808] and the Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM 67 (2020)].