On the hardness of finding the geodetic number of a subcubic graph

A set of vertices D of a graph G is geodetic if every vertex of G lies in a shortest path between two vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G, and deciding whether it is at most k is an NP-complete problem for several classes of graphs. While the problem is easy for graphs of maximum degree at most 2, we show that the problem is NP-complete for graphs of maximum degree three.