We investigate the problem @math of counting all induced subgraphs of size @math in a graph @math that satisfy a given property @math . This continues the work of Jerrum and Meeks who proved the problem to be @math -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 @math , the problem @math is hard for @math if the reduced Euler characteristic of the associated simplicial (graph) complex of @math is non-zero. This observation links @math 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 @math is @math -hard for every monotone property @math 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 @math -edge-connected for @math . Moreover, we show that for those properties @math can not be solved in time @math for any computable function @math unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that @math is @math -hard if @math is any non-trivial modularity constraint on the number of edges with respect to some prime @math or if @math enforces the presence of a fixed isolated subgraph.
Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area. Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of these subgraphs in some sense; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting problem. The paper also gives a thorough survey of known results on this subject and the methods used, as well as discussing the relationships both between multicolour and uncoloured versions of subgraph counting problems, and between exact counting, approximate counting and the corresponding decision problems.