Dynamic programming for H-minor-free graphs

We provide a framework for the design and analysis of dynamic programming algorithms for H-minor-free graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2 ^O(k·logk)·n ^O(1) steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedded in surfaces, we introduce a new type of branch decomposition for H-minor-free graphs, called an H-minor-free cut decomposition, and we show that they can be constructed in O _h(n ³) steps, where the hidden constant depends exclusively on H. We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called ℓ-triangulation. Our main result is that when applied on H-minor-free cut decompositions, dynamic programming runs in 2 ^oh(k)·n ^o(1) steps. This broadens substantially the class of problems that can be solved deterministically in single-exponential time for H-minor-free graphs.