Durham Research Online Deposited in

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 (si, ti) for i = 1, . . . , k and is to test whether G contain k mutually induced paths Pi such that Pi connects si and ti 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.