Parameterizing Path Partitions

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 for short) has been studied extensively, as it includes Hamiltonian Path as a special case. However, the natural variants where the paths are required to be either induced, called Induced Path Partition (IPP for short) or shortest, called Shortest Path Partition (SPP for short), 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. Furthermore, when parameterized by the natural parameter "number of paths", both problems 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 (while 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 when parameterized by the neighborhood diversity of the input graph. Moreover, 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.