  We study the algorithmic complexity of partitioning the vertex set of a given
(di)graph into a small number of paths. The Path Partition problem (PP) has
been studied extensively, as it includes Hamiltonian Path as a special case.
The natural variants where the paths are required to be either \emph{induced}
(Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition,
SPP), have received much less attention. Both problems are known to be
NP-complete on undirected graphs; we strengthen this by showing that they
remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP
remains \NP-hard on undirected bipartite graphs. When parameterized by the
natural parameter ``number of paths'', both SPP and IPP are shown to be
W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected
graphs for the same parameter, as well as for other special subclasses of
directed graphs (IPP is known to be NP-hard on undirected graphs, even for two
paths). On the positive side, we show that for undirected graphs, both problems
are in FPT, parameterized by neighborhood diversity. We also give an explicit
algorithm for the vertex cover parameterization of PP. When considering the
dual parameterization (graph order minus number of paths), all three variants,
IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the
mentioned neighborhood diversity and dual parameterization results to directed
graphs; here, we need to define a proper novel notion of directed neighborhood
diversity. As we also show, most of our results also transfer to the case of
covering by edge-disjoint paths, and purely covering.
