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

Radu Curticapean
2014 arXiv   pre-print
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(n^4) time algorithm for counting perfect matchings in graphs excluding H as a minor. The runtime can be lowered to O(n^1.5) when G excludes K_5 or K_3,3 as a minor. This is the first generalization of an algorithm for counting perfect matchings in K_3,3-free graphs (Little 1974, Vazirani 1989). Our algorithm uses black-boxes for counting perfect matchings in
more » ... r graphs and for computing certain graph decompositions. Together with an independent recent result (Straub et al. 2014) for graphs excluding K_5, it is one of the first nontrivial algorithms to not inherently rely on Pfaffian orientations.
arXiv:1406.4056v1 fatcat:lgd3r75qlnextddcmrdnm47vly