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Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC\u00a017], we discover that for monotone properties\u00a0$$\\varPhi $$<\/jats:tex-math>\n\u03a6<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the problem $$\\#{{\\mathsf {IndSub}}}(\\varPhi )$$<\/jats:tex-math>\n\n#<\/mml:mo>\nIndSub<\/mml:mi>\n(<\/mml:mo>\n\u03a6<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is hard for $$\\#{{\\mathrm {W[1]}}}$$<\/jats:tex-math>\n\n#<\/mml:mo>\n\nW<\/mml:mi>\n[<\/mml:mo>\n1<\/mml:mn>\n]<\/mml:mo>\n<\/mml:mrow>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> if the reduced Euler characteristic of the associated simplicial (graph) complex of $$\\varPhi $$<\/jats:tex-math>\n\u03a6<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is non-zero. This observation links $$\\#{{\\mathsf {IndSub}}}(\\varPhi )$$<\/jats:tex-math>\n\n#<\/mml:mo>\nIndSub<\/mml:mi>\n(<\/mml:mo>\n\u03a6<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> to Karp\u2019s famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the \u201ctopological approach to evasiveness\u201d which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS\u00a083], we prove that $$\\#{{\\mathsf {IndSub}}}(\\varPhi )$$<\/jats:tex-math>\n\n#<\/mml:mo>\nIndSub<\/mml:mi>\n(<\/mml:mo>\n\u03a6<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is\u00a0$$\\#{{\\mathrm {W[1]}}}$$<\/jats:tex-math>\n\n#<\/mml:mo>\n\nW<\/mml:mi>\n[<\/mml:mo>\n1<\/mml:mn>\n]<\/mml:mo>\n<\/mml:mrow>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hard for every monotone property $$\\varPhi $$<\/jats:tex-math>\n\u03a6<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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<\/jats:italic>-edge-connected for $$k > 2$$<\/jats:tex-math>\n\nk<\/mml:mi>\n><\/mml:mo>\n2<\/mml:mn>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Moreover, we show that for those properties $$\\#{{\\mathsf {IndSub}}}(\\varPhi )$$<\/jats:tex-math>\n\n#<\/mml:mo>\nIndSub<\/mml:mi>\n(<\/mml:mo>\n\u03a6<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> can not be solved in time $$f(k)\\cdot n^{o(k)}$$<\/jats:tex-math>\n\nf<\/mml:mi>\n\n(<\/mml:mo>\nk<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n\u00b7<\/mml:mo>\n\nn<\/mml:mi>\n\no<\/mml:mi>\n(<\/mml:mo>\nk<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:msup>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> for any computable function f<\/jats:italic> unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that $$\\#{{\\mathsf {IndSub}}}(\\varPhi )$$<\/jats:tex-math>\n\n#<\/mml:mo>\nIndSub<\/mml:mi>\n(<\/mml:mo>\n\u03a6<\/mml:mi>\n)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$\\#{{\\mathrm {W[1]}}}$$<\/jats:tex-math>\n\n#<\/mml:mo>\n\nW<\/mml:mi>\n[<\/mml:mo>\n1<\/mml:mn>\n]<\/mml:mo>\n<\/mml:mrow>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hard if $$\\varPhi $$<\/jats:tex-math>\n\u03a6<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is any non-trivial modularity constraint on the number of edges with respect to some prime q<\/jats:italic> or if $$\\varPhi $$<\/jats:tex-math>\n\u03a6<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> enforces the presence of a fixed isolated subgraph.<\/jats:p>","DOI":"10.1007\/s00453-020-00676-9","type":"journal-article","created":{"date-parts":[[2020,1,22]],"date-time":"2020-01-22T07:02:50Z","timestamp":1579676570000},"page":"2267-2291","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3159-9418","authenticated-orcid":false,"given":"Marc","family":"Roth","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5774-3508","authenticated-orcid":false,"given":"Johannes","family":"Schmitt","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,1,22]]},"reference":[{"key":"676_CR1","unstructured":"Abrahamson, K.R., Downey, R.G., Fellows, M.R.: Fixed-parameter intractability II (extended abstract). 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