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

We investigate the problem #{IndSub}(Phi) of counting all induced subgraphs of size k in a graph G that satisfy a given property Phi. 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 Phi, the problem #{IndSub}(Phi) is hard for #{W[1]} if the reduced Euler characteristic of the associated simplicial (graph) complex of Phi is non-zero. This observation links #{IndSub}(Phi) 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}(Phi) is #{W[1]}-hard for every monotone property Phi 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}(Phi) 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}(Phi) is #{W[1]}-hard if Phi is any non-trivial modularity constraint on the number of edges with respect to some prime q or if Phi enforces the presence of a fixed isolated subgraph.