  Paths P1,...,Pk in a graph G=(V,E) are said to be mutually induced if for any
1 <= i < j <= k, Pi and Pj have neither common vertices nor adjacent vertices
(except perhaps their end-vertices). The Induced Disjoint Paths problem is to
test whether a graph G with k pairs of specified vertices (si,ti) contains k
mutually induced paths Pi such that Pi connects si and ti for i=1,...,k. We
show that this problem is fixed-parameter tractable for claw-free graphs when
parameterized by k. Several related problems, such as the k-in-a-Path problem,
are proven to be fixed-parameter tractable for claw-free graphs as well. We
show that an improvement of these results in certain directions is unlikely,
for example by noting that the Induced Disjoint Paths problem cannot have a
polynomial kernel for line graphs (a type of claw-free graphs), unless NP
\subseteq coNP/poly. Moreover, the problem becomes NP-complete, even when k=2,
for the more general class of K_1,4-free graphs. Finally, we show that the
n^O(k)-time algorithm of Fiala et al. for testing whether a claw-free graph
contains some k-vertex graph H as a topological induced minor is essentially
optimal by proving that this problem is W[1]-hard even if G and H are line
graphs.
