Clique or hole in claw-free graphs

Given a claw-free graph and two non-adjacent vertices x and y without common neighbours we prove that there exists a hole through x and y unless the graph contains the obvious obstruction, namely a clique separating x and y. We derive two applications: We give a necessary and sufficient condition for the existence of an induced x–z path through y, where x,y,z are prescribed vertices in a claw-free graph; and we prove an induced version of Mengerʼs theorem between four terminal vertices. Finally, we improve the running time for detecting a hole through x and y and for the Three-in-a-Tree problem, if the input graph is claw-free.