  Given an undirected graph $G = (V_G, E_G)$ and a fixed "pattern" graph $H =
(V_H, E_H)$ with $k$ vertices, we consider the $H$-Transversal and $H$-Packing
problems. The former asks to find the smallest $S \subseteq V_G$ such that the
subgraph induced by $V_G \setminus S$ does not have $H$ as a subgraph, and the
latter asks to find the maximum number of pairwise disjoint $k$-subsets $S_1,
..., S_m \subseteq V_G$ such that the subgraph induced by each $S_i$ has $H$ as
a subgraph.
  We prove that if $H$ is 2-connected, $H$-Transversal and $H$-Packing are
almost as hard to approximate as general $k$-Hypergraph Vertex Cover and
$k$-Set Packing, so it is NP-hard to approximate them within a factor of
$\Omega (k)$ and $\widetilde \Omega (k)$ respectively. We also show that there
is a 1-connected $H$ where $H$-Transversal admits an $O(\log k)$-approximation
algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1.
For a special case of $H$-Transversal where $H$ is a (family of) cycles, we
mention the implication of our result to the related Feedback Vertex Set
problem, and give a different hardness proof for directed graphs.
