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We investigate the problem #IndSub(Φ) of counting all induced subgraphs of size k in a graph G that satisfy a given property Φ. This continues the work of Jerrum and Meeks who proved the problem to be #W[1]-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even-or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties Φ, the problem #IndSub(Φ) is hard for #W[1] if the reduced Euler characteristic of the associated simplicial (graph) complex of Φ is nonzero. This observation links #IndSub(Φ) to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that #IndSub(Φ) is #W[1]-hard for every monotone property Φ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for k > 2. Moreover, we show that for those properties #IndSub(Φ) can not be solved in time f (k) · n o(k) for any computable function f unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that #IndSub(Φ) is #W[1]-hard if Φ is any non-trivial modularity constraint on the number of edges with respect to some prime q or if Φ enforces the presence of a fixed isolated subgraph.

Jaeger , Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #Phard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y = 1, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp o(n) unless #ETH fails.Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp o(n) unless #ETH fails.In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation.

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