  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
(except perhaps their end-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 $P_i$ connects $s_i$ and $t_i$.
This is a classical graph problem that is NP-complete even for $k=2$. We study
it for AT-free graphs.
  Unlike its subclasses of permutation graphs and cocomparability graphs, the
class of AT-free graphs has no geometric intersection model. However, by a new,
structural analysis of the behaviour of Induced Disjoint Paths for AT-free
graphs, we prove that it can be solved in polynomial time for AT-free graphs
even when $k$ is part of the input. This is in contrast to the situation for
other well-known graph classes, such as planar graphs, claw-free graphs, or
more recently, (theta,wheel)-free graphs, for which such a result only holds if
$k$ is fixed.
  As a consequence of our main result, the problem of deciding if a given
AT-free graph contains a fixed graph $H$ as an induced topological minor admits
a polynomial-time algorithm. In addition, we show that such an algorithm is
essentially optimal by proving that the problem is W[1]-hard with parameter
$|V_H|$, even on a subclass of AT-free graph, namely cobipartite graphs. We
also show that the problems $k$-in-a-Path and $k$-in-a-Tree are polynomial-time
solvable on AT-free graphs even if $k$ is part of the input. These problems are
to test if a graph has an induced path or induced tree, respectively, spanning
$k$ given vertices.
