Cardinality constrained path covering problems in grid graphs

In this article we continue our study on the complexity of Path Covering Problems started in [2]. Here, taking one further step, we investigate the complexity of the problem on grids. For special classes of grids (general grids, grids with a fixed number of rows, ladders), and several special unweighted path collections (general paths, paths of length 2, L-shaped paths, pipes, hooks, staples) we either give polynomial-time algorithms or prove NP-completeness results