Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $\Phi$, the problem $\#\ensuremath{{\sc IndSub}}(\Phi)$ asks, on input of $G$ and positive integer $k$, to compute number $\#{\ensuremath{{IndSub}({\Phi,k} \to {G})}}$ induced subgraphs size $k$ in that satisfy $\Phi$. The search for explicit criteria $\Phi$ ensuring is hard was initiated by Jerrum Meeks [J. Comput. System Sci., 81 (2015), pp. 702--716] part major line research counting small patterns graphs. However, apart from an implicit result due Curticapean, Dell, Marx [STOC, ACM, New York, 151--158] proving full classification into “easy” “hard” properties possible some partial results edge-monotone [Discrete Appl. Math., 198 (2016), 170--194] Dörfler et al. [MFCS, LIPIcs Leibniz Int. Proc. Inform. 138, Wadern Germany, 2019, 26], not much known. In this work, we fully answer explicitly classify case monotone, is, subgraph-closed, properties: We show any nontrivial monotone cannot be solved time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ function $f$, unless exponential hypothesis fails. By this, establish significant improvement over brute-force approach unlikely; language parameterized complexity, also obtain $\#{\ensuremath{{W[1]}}}$-completeness result. methods develop above allow us prove conjecture [ACM Trans. Theory, 7 11; Combinatorica 37 (2017), 965--990]: $\#{\ensuremath{{W[1]}}}$-complete if only depending edges graph.