Counting perfect matchings in graphs that exclude a single-crossing minor

A graph H is single-crossing if it can be drawn in the plane with at most one crossing. For any single-crossing graph H, we give an O(n4) time algorithm for counting perfect matchings in graphs excluding H as a minor. The runtime can be lowered to O(n1.5) when G excludes K5 or K3,3 as a minor. This is the first generalization of an algorithm for counting perfect matchings in K3,3free graphs (Little 1974, Vazirani 1989). Our algorithm uses black-boxes for counting perfect matchings in planar graphs and for computing certain graph decompositions. Together with an independent recent result (Straub et al. 2014) for graphs excluding K5, it is one of the first nontrivial algorithms to not inherently rely on Pfaffian orientations.