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Simple, Strict, Proper, and Directed: Comparing Reachability in Directed and Undirected Temporal Graphs

Author Michelle Döring

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LIPIcs.ISAAC.2025.27.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Paths and connectivity problems
  • Networks → Network dynamics
Keywords
  • temporal graphs
  • directed graphs
  • temporal reachability
  • dynamic networks

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Abstract

Temporal graphs model networks whose connections are available only at specific points in time. Several definitional subtleties - whether paths must follow strictly increasing time labels (strict vs. non-strict), whether adjacent edges cannot appear simultaneously (proper), and whether edges are forbidden to appear multiple times (simple) - give rise to different temporal graph settings. These distinctions directly impact the definition of temporal reachability, a core concept in temporal graph theory. Casteigts, Corsini, and Sarkar [TCS24] introduced a framework of equivalence notions to compare the expressive power of these settings focusing solely on undirected temporal graphs. In this work, we extend their framework to include the fundamental dimension of directed vs. undirected.
Our contribution is three-fold. We (1) complete the undirected hierarchy by resolving the two open questions from [TCS24], (2) fully characterize the hierarchy of the directed settings, and (3) compare the directed and undirected settings, showing that directed temporal graphs are strictly more expressive than undirected temporal graphs in terms of reachability.
Our structural results highlight both the limitations and strengths of various temporal graph settings - for example, directed + strict + simple graphs can realize every possible reachability graph, while directed + proper graphs necessarily induce at least one transitive reachability on each directed cycle. We also provide transformation procedures between temporal settings offering practical tools for transferring algorithms and hardness results across models.

Cite As Get BibTex

Michelle Döring. Simple, Strict, Proper, and Directed: Comparing Reachability in Directed and Undirected Temporal Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.27

Author Details

Michelle Döring
  • Hasso Plattner Institute, University of Potsdam, Germany

Funding

  • Döring, Michelle: German Federal Ministry for Education and Research (BMBF) through the project "KI Servicezentrum Berlin Brandenburg" (01IS22092).

Acknowledgements

I warmly thank Arnaud Casteigts for his advice and inspiring discussions, and George Skretas for his steady support, encouragement, and valuable feedback on this work.

References

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