Some remarks on the geodetic number of a graph

A set of vertices DD of a graph GG is geodetic if every vertex of GG lies on a shortest path between two not necessarily distinct vertices in DD. The geodetic number of GG is the minimum cardinality of a geodetic set of GG.We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph GG and a given integer kk whether GG has a geodetic set of cardinality at most kk. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.