Counting the Number of Matchings in Chordal and Chordal Bipartite Graph Classes

摘要

We provide polynomial-time algorithms for counting the number of perfect matchings in chain graphs, cochain graphs, and threshold graphs. These algorithms are based on newly developed subdivision schemes that we call a recursive decomposition. On the other hand, we show the #P-completeness for counting the number of perfect matchings in chordal graphs, split graphs and chordal bipartite graphs. This is in an interesting contrast with the fact that counting the number of independent sets in chordal graphs can be done in linear time.