Parameterizing Path Partitions

Abstract

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 induced (Induced Path Partition, IPP) or shortest (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be $$\textsf {NP}$$ -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 $$\textsf {NP}$$ -hard on undirected bipartite graphs. When parameterized by the natural parameter “number of paths”, both problems are shown to be $$\textsf {W}[1]$$ -hard on DAGs. We also show that SPP is in $$\textsf {XP}$$ both for DAGs and undirected graphs for the same parameter (IPP is known to be $$\textsf {NP}$$ -hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in $$\textsf {FPT}$$ , parameterized by neighborhood diversity. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in $$\textsf {FPT}$$ for undirected graphs.