Partitioning vertices of graphs into paths of the same length

Abstract

Given a graph, the (induced) $P_k$-partition problem is to decide whether its vertex set can be partitioned into subsets, each of which induces (the $k$-path) a $k$-vertex subgraph with a Hamiltonian path. We show that these problems are NP-complete for planar subcubic bipartite $(H_1,H_2,\dots ,H_{\ell })$-free graphs of girth $g$, for any $k,g\ge 3,l\ge 1$, where $H_i$ is obtained by joining central vertices in two copies of $P_3$ with $P_{i+1}$. We show that the $P_k$-partition (induced $P_k$-partition) problem is NP-complete for split graphs and any $k\ge 5$, chordal graphs and any $k\ge 4$ (any $k\ge 3$), line graphs of planar bipartite graphs and any $k\ge 5$ (any $k\ge 3$). We show that the $P_4$-partition and, for any $k\ge 5$, induced $P_k$-partition problems, restricted to split graphs, are polynomial. Additionally, we prove NP-completeness for the optimization version of the induced $P_4$-partition problem and split graphs.