@article{22812,
  abstract     = {Efficient data structures for computing the cutset of a set of nodes in a graph undergoing dynamic edge insertions/deletions are well-studied in undirected graphs. We study this problem in directed graphs and show a reduction from the Online Boolean Matrix-Vector Multiplication Conjecture (OMv) introduced by Henzinger et al. [STOC’15]. We prove conditional on OMv that a dynamic data structure computing the directed cutset of a set of nodes cannot have an amortized time of both O(n2−ε) for a query operation and O(n1−ε) for an update operation, for any constant ε > 0, even when an adversary’s operations are restricted to the incremental or decremental settings. We further give algorithms to match these lower bounds.},
  author       = {Hahn, Niklas and Henzinger, Monika H and Stefankovic, Zofia},
  issn         = {0020-0190},
  journal      = {Information Processing Letters},
  keywords     = {Dynamic algorithms, Graph algorithms, Directed graphs, Lower bounds, Upper bounds},
  publisher    = {Elsevier},
  title        = {{Tight bounds on the performance of dynamic directed cutset data structures based on OMv}},
  doi          = {10.1016/j.ipl.2026.106663},
  volume       = {195},
  year         = {2026},
}

@inproceedings{22918,
  abstract     = {Given a weighted digraph G, a (t,g,μ)-DAG cover is a collection of g dominating DAGs D_1,… ,D_g such that all distances are approximately preserved: for every pair (u,v) of vertices, min_id_{D_i}(u,v) ≤ t⋅ d_G(u,v), and the total number of non-G edges is bounded by |(∪_i D_i)⧵ G| ≤ μ. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of Steiner DAG cover, where the DAGs are allowed to contain Steiner points. 
We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a (1,2,Õ(n⋅tw))-Steiner DAG cover. For planar digraphs we provide a (1+ε,2,Õ_ε(n))-Steiner DAG cover.
We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth 1 with stretch t < 2 and sub-quadratic number of extra edges requires Ω(log n) DAGs.},
  author       = {Bhore, Sujoy and Chang, Hsien Chih and Conroy, Jonathan and Filtser, Arnold and Oh, Eunjin and Wein, Nicole and Zheng, Da Wei},
  booktitle    = {34th Annual European Symposium on Algorithms},
  isbn         = {9783959774451},
  issn         = {1868-8969},
  keywords     = {Directed graphs, DAG (directed acyclic graphs), distortion, metric embeddings, planar graph, treewidth},
  location     = {L’Aquila, Italy},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{DAG covers for structured graphs: The Steiner point effect}},
  doi          = {10.4230/LIPIcs.ESA.2026.94},
  volume       = {388},
  year         = {2026},
}