Killing a Vortex

The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph $H,$ every $H$-minor-free graph can be obtained by clique-sums of ``almost embeddable'' graphs. Here a graph is ``almost embeddable'' if it can be obtained from a graph of bounded Euler-genus by pasting graphs of bounded pathwidth in an ``orderly fashion'' into a bounded number of faces, called the \textit{vortices}, and then adding a bounded number of additional vertices, called \textit{apices}, with arbitrary neighborhoods. Our main result is a {full classification} of all graphs $H$ for which the use of vortices in the theorem above can be avoided. To this end we identify a (parametric) graph $\mathscr{S}_{t}$ and prove that all $\mathscr{S}_{t}$-minor-free graphs can be obtained by clique-sums of graphs embeddable in a surface of bounded Euler-genus after deleting a bounded number of vertices. We show that this result is tight in the sense that the appearance of vortices cannot be avoided for $H$-minor-free graphs, whenever $H$ is not a minor of $\mathscr{S}_{t}$ for some $t\in\mathbb{N}.$ Using our new structure theorem, we design an algorithm that, given an $\mathscr{S}_{t}$-minor-free graph $G,$ computes the generating function of all perfect matchings of $G$ in polynomial time. Our results, combined with known complexity results, imply a complete characterization of minor-closed graph classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every $\mathscr{S}_{t}$ as a minor. This provides a \textit{sharp} complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.