NC Algorithms for Computing the Number of Perfect Matchings in K3, 3-free Graphs and Related Problems

We show that the problem of computing the number of perfect matchings in K3,3-free graphs is in NC. This stands in striking contrast with the #P-completeness of counting the number of perfect matchings in arbitrary graphs. As corollaries we obtain NC algorithms for checking if a given K3,3-free graph has a perfect matching and if it has an EXACT MATCHING. Our result also opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K3,3-free graphs.

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