It is known that determinizing a nondeterministic input-driven pushdown automaton (NIDPDA) of size n results in the worst case in a machine of size (R. Alur, P. Madhusudan, "Adding nesting structure to words", J.ACM 56(3), 2009). This paper considers the special case of k-path NIDPDAs, which have at most k computations on any input. It is shown that the smallest deterministic IDPDA equivalent to a k-path NIDPDA of size n is of size Θ(n ^k ). The paper also gives an algorithm for deciding whether or not a given NIDPDA has the k-path property, for a given k; if k is fixed, the problem is P-complete.