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.

翻译：给定图G和整数k，Π-收缩问题的目标是检查G中是否存在不超过k条边，使得收缩这些边后得到的图满足性质Π。我们研究Π为"无H"（不包含任何诱导子图H）的情形。当H是至多两个顶点的完全图时，无H收缩问题显然可在多项式时间内解决。我们证明，在所有其他情况下，该问题是NP完全的。随后我们研究这些问题的固定参数可解性。我们证明，当H是树时（除七种树外），无H收缩问题是W[2]-困难的。这一结果与已知结论共同使得树中仅剩三种情况尚未明确。另一方面，我们取得积极结果：当H为爪形图时，该问题是固定参数可解的。