Vertex-ordering and arc-partitioning problems

We study vertex-ordering problems in loop-free digraphs subject to constraints on the left-going arcs, focusing on existence conditions and computational complexity. As an intriguing special case, we explore vertex-specific lower and upper bounds on the left-outdegrees and right-indegrees. We show, for example, that deciding whether the left-going arcs can form an in-branching is solvable in polynomial time and provide a necessary and sufficient condition, while the analogous problem for an in-arborescence turns out to be NP-complete. We also consider a weighted variant that enforces vertex-specific lower and upper bounds on the weighted left-outdegrees, which is particularly relevant in applications. Furthermore, we investigate the connection between ordering problems and their arc-partitioning counterparts, where one seeks to partition the arcs into a subgraph from a specific digraph family and an acyclic subgraph -- equivalently, one seeks to cover all directed cycles with a subgraph belonging to a specific family. For the family of in-branchings, unions of disjoint dipaths, and matchings, the two formulations coincide, whereas for in-arborescences, dipaths, Hamiltonian dipaths, and perfect matchings the formulations diverge. Our results yield a comprehensive complexity landscape, unify diverse special cases and variants, clarify the algorithmic boundaries of ordered digraphs, and relate them to broader topics including graph degeneracy, acyclic orientations, influence propagation, and rank aggregation.