Under the temporal Dilworth promise, minimum temporal (disjoint) path covers equal the Lovász number of the connectivity graph and are poly-time solvable; TPC is W[1]-hard by deletion distance to linear forest but FPT by vertex cover when time-steps are parameterized.
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Temporal Path Covers: Dilworth Properties and Parameterized Complexity
Under the temporal Dilworth promise, minimum temporal (disjoint) path covers equal the Lovász number of the connectivity graph and are poly-time solvable; TPC is W[1]-hard by deletion distance to linear forest but FPT by vertex cover when time-steps are parameterized.