Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that lpt(G) \leq ceiling(n/4-n^{2/3}/90), if G is connected, lct(G) \leq ceiling(n/3-n^{2/3}/36), if G is 2-connected, and \lpt(G) \leq 3, if G is a connected circular arc graph. Our bound on lct(G) improves an earlier result of Thomassen and our bound for circular arc graphs relates to an earlier statement of Balister \emph{et al.} the argument of which contains a gap. Furthermore, we prove upper bounds on lpt(G) for planar graphs and graphs of bounded tree-width.