Geodetic sets for directed acyclic planar geodetic graphs

A set of vertices $S$ of a directed graph G is {\em geodetic} if every vertex of G lies on a shortest path from a vertex of S to a vertex of S. A directed graph is geodetic if there is at most one shortest path from every vertex of G to every vertex of G. We prove the NP-completeness of the following decision problem. Given a directed acyclic planar geodetic graph G and an integer k, does G have a geodetic set with at most k vertices? This implies that the question of whether G has a strong or a monitoring geodetic set with at most k vertices is also NP-complete for directed acyclic planar geodetic graphs. Furthermore, it is shown that the number of vertices in a minimum geodetic set and the number of vertices in a minimum edge geodetic set can be computed in linear time for directed acyclic series-parallel graphs.