Counting the Number of Perfect Matchings in K5-Free Graphs

Counting the number of perfect matchings in arbitrary graphs is a sharpP-complete problem. However, for some restricted classes of graphs the problem can be solved efficiently. In the case of planar graphs, and even for K_{3, 3}-free graphs, Vazirani showed that it is in NC^2. The technique there is to compute a Pfaffian orientation of a graph. In the case of K_5-free graphs, this technique will not work because some K_5-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K_5-free graphs can be computed in polynomial time and we describe a circuit construction in TC2.