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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 counting 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/logk) 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 TGk of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, TGk(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 TGk at every pair of rational coordinates (x,y)
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