Abstract
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,aEuro broken vertical bar,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,aEuro broken vertical bar,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.