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.

No abstract

No abstract

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