Catalan structures and dynamic programming in H-minor-free graphs

We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex H-minor-free graph G contains a path of length k in 2 ^O(k)nO(¹⁾ steps. Our approach builds on a combination of Demaine-Hajiaghayis bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of H-minor-free graphs. This result is used to bound the number of ways vertex disjoint paths can be routed through the separators of such decompositions. The proof is based on several structural theorems from the Graph Minors series of Robertson and Seymour. With a slight modification, similar combinatorial and algorithmic results can be derived for many other problems. Our approach can be viewed as a general framework for obtaining time 2^O(k)nO(¹⁾ algorithms on H-minor-free graph classes.