Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

We investigate the problem $$ of counting all induced subgraphs of size k in a graph G that satisfy a given property $$ . This continues the work of Jerrum and Meeks who proved the problem to be $$ -hard for some families of properties which include (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties  $$ , the problem $$ is hard for $$ if the reduced Euler characteristic of the associated simplicial (graph) complex of $$ is non-zero. This observation links $$ 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 $$ is  $$ -hard for every monotone property $$ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for $$ . Moreover, we show that for those properties $$ can not be solved in time $$ 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 $$ is $$ -hard if $$ is any non-trivial modularity constraint on the number of edges with respect to some prime q or if $$ enforces the presence of a fixed isolated subgraph.