The Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) problems were introduced by [Chakraborty, Dailly, Foucaud, Klasing, MFCS '24] as the natural temporal analogues of Minimum Path Cover and Minimum Path Partition, and were shown to be NP-hard on temporal DAGs. All tractable graph classes for T(D)PC established in that paper satisfy a temporal Dilworth property, namely that the size of the respective minimum temporal path cover is equal to the size of the maximum antichain. This raises a natural question: is T(D)PC guaranteed to be polynomial-time solvable under the promise that the respective Dilworth property holds? For both problems, we answer this question in the affirmative for their decision version and in the negative for their search version (under P $\neq$ NP). In another direction, we establish parameterized algorithms and hardness results for TPC and TDPC. Our main result is that TPC is W[1]-hard parameterized by the deletion distance to linear forest even for temporal graphs with two time-steps, answering in the negative an open question by Chakraborty et al. about whether an XP algorithm parameterized by treewidth plus number of time-steps can be improved to FPT. On the other hand, we prove that an FPT algorithm does exist if the vertex cover number is used as parameter instead of the treewidth in the above parameterization. We complement this with a proof that including the number of time-steps in the parameter is necessary to yield tractability, as, otherwise, both TPC and TDPC remain NP-hard even for constant vertex cover size. Along the way, we establish various other para-NP-hardness results involving structural parameters such as the pathwidth and the maximum degree of the underlying graph.

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