Contracting edges to destroy a pattern: A complexity study

Given a graph G and an integer k, the objective of the $\Pi$-Contraction problem is to check whether there exists at most k edges in G such that contracting them in G results in a graph satisfying the property $\Pi$. We investigate the problem where $\Pi$ is `H-free' (without any induced copies of H). It is trivial that H-free Contraction is polynomial-time solvable if H is a complete graph of at most two vertices. We prove that, in all other cases, the problem is NP-complete. We then investigate the fixed-parameter tractability of these problems. We prove that whenever H is a tree, except for seven trees, H-free Contraction is W[2]-hard. This result along with the known results leaves behind three unknown cases among trees. On a positive note, we obtain that the problem is fixed-parameter tractable, when H is a paw.