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 2o(k)⋅∣G∣O(1), unless Exponential Time Hypothesis fails.
@article{arxiv.1507.06341,
title = {Parameterized lower bound and NP-completeness of some $H$-free Edge Deletion problems},
author = {N. R. Aravind and R. B. Sandeep and Naveen Sivadasan},
journal= {arXiv preprint arXiv:1507.06341},
year = {2015}
}