Parameterized Complexity of Geodetic Set

A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb N$, Geodetic Set asks whether there is a geodetic set of size at most $k$. We study the parameterized complexity of Geodetic Set with respect to structural parameters and show dichotomy results: We develop fixed-parameter algorithms with respect to the feedback edge number and with respect to the tree-depth. On the negative side, we prove that the problem is W[1]-hard with respect to the feedback vertex number and the path-width.