The next-to-shortest path problem on directed graphs with positive edge weights

Given an edge-weighted graph G and two distinct vertices s and t of G, the next-to-shortest path problem asks for a path from s to t of minimum length among all paths from s to t except the shortest ones. In this article, we consider the version where G is directed and all edge weights are positive. Some properties of the requested path are derived when G is an arbitrary digraph. In addition, if G is planar, an O (n 3)-time algorithm is proposed, where n is the number of vertices of G.