  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 $k \in \mathbb
N$, the NP-hard Geodetic Set problem asks whether there is a geodetic set of
size at most $k$. Complementing various works on Geodetic Set restricted to
special graph classes, we initiate a parameterized complexity study of Geodetic
Set and show, on the negative side, that Geodetic Set is W[1]-hard when
parameterized by feedback vertex number, path-width, and solution size,
combined. On the positive side, we develop fixed-parameter algorithms with
respect to the feedback edge number, the tree-depth, and the modular-width of
the input graph.
