Counting the Number of Perfect Matchings in K5-free Graphs

Simon Straub, Thomas Thierauf, and Fabian Wagner

Counting the number of perfect matchings in arbitrary graphs is a #P-complete problem. However, for some restricted classes of graphs the problem can be solved efficiently. In the case of planar graphs, and even for K3,3-free graphs, Vazirani showed that it is in NC2. The technique there is to compute a Pfaffian orientation of a graph.

In the case of K5-free graphs, this technique will not work because some K5-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K5-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC2.