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A proper coloring of a graph is a labeled partition of its vertices into parts which are independent sets.

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Abstract. 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. We prove that H-free Edge Deletion is NP-complete if H is a graph with at least two edges and H has a component with maximum number of vertices which is a tree or a regular graph. Furthermore, we obtain that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time 2 o(k) · |G| O(1) , unless Exponential Time Hypothesis fails.