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, the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs. 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, 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 $O(2^s \cdot n^3)$ time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an $f(k,s)\cdot n^{O(1)}$ time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces.