The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every there exists some constant such that every -minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most and “at most vortices of depth ”. Our main combinatorial result is a “vortex-free” refinement of the above structural theorem as follows: we identify a (parameterized) graph , called shallow vortex grid, and we prove that if in the above structural theorem we replace by then the resulting decomposition becomes “vortex-free”. Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an -minor-free graph , computes the generating function of all perfect matchings of in polynomial time. This algorithm yields, on -minor-free graphs, polynomial algorithms for computational problems such as the {dimer problem, the exact matching problem}, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.
This is joint work with Dimitrios M. Thilikos.