We introduce graph motif parameters, 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 H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies ofby a surprisingly simple algorithm. This improves upon previously known running times, such as O(n 0.91k+c ) time for k-edge matchings or O(n 0.46k+c ) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ∈ C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds.Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy:

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 corresponding lower bound: if H is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub(H) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT = #W [1].As a first step of the proof, we show that counting k-matchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] [16] proved the #W[1]-hardness of counting k-matchings in general graphs; our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f (k)n o(k/ log k) time algorithm for counting k-matchings in bipartite graphs for any computable function f (k). As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] [23] stating that counting paths of length k is #W[1]-hard, as well as a similar almosttight ETH-based lower bound on the exponent.

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We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s, t)-path in G that is at least k longer than a shortest (s, t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) · poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k.Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s, t)-path that is exactly k longer than a shortest (s, t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746 k · poly(n), and a deterministic algorithm with running time about 6.745 k · poly(n), showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs. * Extended abstract appears at ICALP 2017. Proposition 2 ([5, 16]).There is an algorithm with running time 2 O(tw(G)) · n O(1) that computes a longest path between two given vertices of a given graph.Let us note that the running time of Proposition 2 can be improved to 2 O(tw(G)) · n by making use of the matroid-based approach from [16].Our main theorem is based on graph minors, and we introduce some notation here. Definition 3. A topological minor model of H in

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 structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30 n ) and at least Ω(2.43 n ) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by n O(k) , where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414 n ). Given that there are several well-studied configurations of points with at least Ω(3.464 n ) triangulations, and some even with Ω(8 n ) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464 n ) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W [2]-hard in the parameter k. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

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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 minor).To study these parameters, we first introduce the notion of combined matchgates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and matchgates: Using combined matchgates, we can simulate certain non-existing gadgets F as linear combinations of t = O(1) existing gadgets. If a graph G features k occurrences of F , we can then reduce G to t k graphs that feature only existing gadgets, thus enabling parameterized reductions.As applications of this technique, we simplify known 4 g n O(1) time algorithms for PerfMatch on graphs of genus g. Orthogonally to this, we show #W[1]-hardness of the permanent on k-apex graphs, implying its #W[1]-hardness under the Hadwiger number. Additionally, we rule out n o(k/ log k) time algorithms under the counting exponential-time hypothesis #ETH.Finally, we use combined matchgates to prove ⊕W[1]-hardness of evaluating the permanent modulo 2 k , complementing an O(n 4k−3 ) time algorithm by Valiant and answering an open question of Björklund. We also obtain a lower bound of n Ω(k/ log k) under the parity version ⊕ETH of the exponential-time hypothesis.