Killing a Vortex
share
The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every t∈ℕ, there exists some constant c_t such that every K_t-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most c_t vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most c_t and "at most c_t vortices of depth c_t". Our main combinatorial result is a "vortex-free" refinement of the above structural theorem as follows: we identify a (parameterized) graph H_t, called shallow vortex grid, and we prove that if in the above structural theorem we replace K_t by H_t, 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 H_t, the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an H_t-minor-free graph G, computes the generating function of all perfect matchings of G in polynomial time. This algorithm yields, on H_t-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 H_t as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.
READ FULL TEXT