For a graph H, the H-free Edge Deletion problem asks whether there exist at most k edges whose deletion from the input graph G results in a graph without any induced copy of H. H-free Edge Completion and H-free Edge Editing are defined similarly where only completion (addition) of edges are allowed in the former and both completion and deletion are allowed in the latter. We completely settle the classical complexities of these problems by proving that H-free Edge Deletion is NP-complete if and only if H is a graph with at least two edges, H-free Edge Completion is NP-complete if and only if H is a graph with at least two non-edges and H-free Edge Editing is NP-complete if and only if H is a graph with at least three vertices. Our result on H-free Edge Editing resolves a conjecture by Alon and Stav (2009). Additionally, we prove that, these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time 2o(k) · |G|O(1), unless Exponential Time Hypothesis fails. Furthermore, we obtain implications on the incompressibility of these problems. © Springer-Verlag Berlin Heidelberg 2016.