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This is a Quantitative Study study.

October 3, 2025Open Access

Vertex-ordering and arc-partitioning problems

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Deciding if left-going arcs form an in-branching is solvable in polynomial time, while the in-arborescence case is NP-complete.
The weighted variant imposes vertex-specific bounds on left-outdegrees, enhancing its relevance in real-world applications.
The study clarifies algorithmic boundaries in ordered digraphs by documenting a rich complexity landscape across various formulations.
Connections between ordering problems and arc-partitioning counterparts reveal significant insights into graph theory applications.

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Analysis reveals polynomial time solutions for in-branching in vertex-ordering problems, indicating complex relationships with arc-partitioning.

Borsik et al. (2025) studied this question.

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Analysis reveals polynomial time solutions for in-branching in vertex-ordering problems, indicating complex relationships with arc-partitioning.