The complexity of finding maximum disjoint paths with length constraints

The following problem is considered: Given an integer K, a graph G with two distinct vertices s and t, find the maximum number of disjoint paths of length K from s to t. The problem has several variants: the paths may be vertex‐disjoint or edge‐disjoint, the lengths of the paths may be equal to K or bounded by K, the graph may be undirected or directed. It is shown that except for small values of K all the problems are NP‐complete. Assuming P ≠ NP, for each problem, the largest value of K for which the problem is not NP‐complete is found. Whenever a polynomial algorithm exists, an efficient algorithm is described.