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For a graph $H$, the $H$-free Edge Deletion problem asks whether there existat most $k$ edges whose deletion from the input graph $G$ results in a graphwithout any induced copy of $H$. We prove that $H$-free Edge Deletion isNP-complete if $H$ is a graph with at least two edges and $H$ has a componentwith maximum number of vertices which is a tree or a regular graph.Furthermore, we obtain that these NP-complete problems cannot be solved inparameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$,unless Exponential Time Hypothesis fails.

Parameterized lower bound and NP-completeness of some $H$-free Edge Deletion problems