Given a graph property Φ, we consider the problem 𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that satisfies Φ. Specifically, we study the parameterized complexity of 𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) and of its counting problem #𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties Φ: the decision problem 𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) always admits an FPT algorithm and the

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... unting problem #𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property Φ which, if satisfied, yields fixed-parameter tractability and otherwise #𝖶[1]-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for #𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) that run in time f(k)·|G|^o(k/log k) for any computable function f. As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ). This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial T^k_G of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, T^k_G(2,1) corresponds to the number of k-forests in the graph G. Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of T^k_G at every pair of rational coordinates (x,y).