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We prove #W[1]-hardness of the following parameterized counting problem: Given a simple undirected graph G and a parameter k ∈ N, compute the number of matchings of size k in G. It is known from [1] that, given an edge-weighted graph G, computing a particular weighted sum over the matchings in G is #W[1]-hard. In the present paper, we exhibit a reduction… (More)

For a class H of graphs, #Sub(H) is the counting problem that, given a graph H ∈ H and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if H has bounded vertex-cover number (equivalently, the size of the maximum matching in H is bounded), then #Sub(H) is polynomial-time solvable. We complement this result with a… (More)

We devise a framework for proving tight lower bounds under the counting exponential-time hypothesis #ETH introduced by Dell et al. (ACM Transactions on Algorithms, 2014). Our framework allows to convert many known #P-hardness results for counting problems into tight lower bounds of the following type: Assuming #ETH, the given problem admits no algorithm… (More)

- Radu Curticapean
- 2015

Let P be a set of $n$ points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such… (More)

Let P be a set of n points in the plane. A crossing-free structure on P is a plane graph with vertex set P. Examples of crossing-free structures include trian-gulations of P , spanning cycles of P , also known as polygonalizations of P , among others. There has been a large amount of research trying to bound the number of such structures. In particular,… (More)

We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph G. These generalize the well-known tractable planar case, and they include the genus of G, its apex number (the minimum number of vertices whose removal renders G planar), and its Hadwiger number (the size of a largest clique… (More)

A graph H is single-crossing if it can be drawn in the plane with at most one crossing. For any single-crossing graph H, we give an O(n 4) time algorithm for counting perfect matchings in graphs excluding H as a minor. The runtime can be lowered to O(n 1.5) when G excludes K 5 or K 3,3 as a minor. This is the first generalization of an algorithm for… (More)