  Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two
distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices.
The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs
of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$
such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical
graph problem that is NP-complete even for $k=2$. We introduce a natural
generalization, Induced Disjoint Connected Subgraphs: instead of connecting
pairs of terminals, we must connect sets of terminals. We give almost-complete
dichotomies of the computational complexity of both problems for H-free graphs,
that is, graphs that do not contain some fixed graph H as an induced subgraph.
Finally, we give a complete classification of the complexity of the second
problem if the number k of terminal sets is fixed, that is, not part of the
input.
