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For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomial-time solvable. We complement this result with(More)
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)
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)
We introduce <em>graph motif parameters</em>, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs <i>H</i>, the number of <i>H</i>-copies (induced or not) in an input graph <i>G</i>, and the number of(More)
The complexity of approximately counting independent sets in bipartite graphs (#BIS) is a central open problem in approximate counting, and it is widely believed to be neither easy nor NP-hard. We study several natural parameterised variants of #BIS, both from the polynomial-time and from the fixed-parameter viewpoint: counting independent sets of a given(More)
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)