The k‐in‐a‐Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP‐complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw‐free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k‐Induced Disjoint Paths and the k‐in‐a‐Cycle problem are solvable in polynomial time on claw‐free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s
i
,t
i
) for i=1,…,k and is to test whether G contain k mutually induced paths P
i
such that P
i
connects s
i
and t
i
for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP‐complete, even for the class of line graphs, which form a subclass of the class of claw‐free graphs.
