Abstract: The algorithms for finding the longest path (the shortest path) and the longest elementary path (the shortest elementary path) with an exact number of arcs from one vertex to each of the vertices on a strongly weighted directed graph are given. The effectiveness of the algorithms is discussed. The algorithms have concrete significance for developing Karp's and Cohen's conclusions which are the algorithms for finding both the minimum cycle mean on a strongly weighted graph and the eigenvalue of the closed loop system matrix of the linear discrete event system in the sense of max algebra.

Key words: optimal control, aeolotropism, closed cycles