Counting the Number of Perfect Matchings in K5-Free Graphs

Abstract

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 K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3,3</sub> -free graphs, Vazirani showed that it is in NC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The technique there is to compute a Pfaffian orientation of a graph. In the case of K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> -free graphs, this technique will not work because some K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> -free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> -free graphs can be computed in polynomial time and we describe a circuit construction in TC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .