Counting Matchings with k Unmatched Vertices in Planar Graphs

We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm [26, 33, 27], the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs [23]. To interpolate between the hard case of counting matchings and the easy case of counting perfect matchings, we study the parameterized problem of counting matchings with exactly k unmatched vertices in a planar graph G, on input G and k. This setting has a natural interpretation in statistical physics, and it is a special case of counting perfect matchings in k-apex graphs (graphs that can be turned planar by removing at most k vertices). Starting from a recent #W[1]-hardness proof for counting perfect matchings on k-apex graphs [12], we obtain that counting matchings with k unmatched vertices in planar graphs is #W[1]-hard. In contrast, given a plane graph G with s distinguished faces, there is an O(2 · n3) time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an f(k, s) · nO(1) time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces.