Enumeration of s-d Separators in DAGs with Application to Reliability Analysis in Temporal Graphs

Authors Alessio Conte, Pierluigi Crescenzi, Andrea Marino, Giulia Punzi

Thumbnail PDF


  • Filesize: 0.68 MB
  • 14 pages

Document Identifiers

Author Details

Alessio Conte
  • Università degli Studi di Pisa, Dipartimento di Informatica, Italy
Pierluigi Crescenzi
  • Université de Paris, IRIF, CNRS, France
  • On-leave from Università degli Studi di Firenze, DiMaI, Firenze, Italy
Andrea Marino
  • Università degli Studi di Firenze, DiSIA, Firenze, Italy
Giulia Punzi
  • Università degli Studi di Pisa, Dipartimento di Informatica, Italy


We would like to thank Gaurav Khanna and Lhouari Nourine for several useful discussion by e-mail.

Cite AsGet BibTex

Alessio Conte, Pierluigi Crescenzi, Andrea Marino, and Giulia Punzi. Enumeration of s-d Separators in DAGs with Application to Reliability Analysis in Temporal Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Temporal graphs are graphs in which arcs have temporal labels, specifying at which time they can be traversed. Motivated by recent results concerning the reliability analysis of a temporal graph through the enumeration of minimal cutsets in the corresponding line graph, in this paper we attack the problem of enumerating minimal s-d separators in s-d directed acyclic graphs (in short, s-d DAGs), also known as 2-terminal DAGs or s-t digraphs. Our main result is an algorithm for enumerating all the minimal s-d separators in a DAG with O(nm) delay, where n and m are respectively the number of nodes and arcs, and the delay is the time between the output of two consecutive solutions. To this aim, we give a characterization of the minimal s-d separators in a DAG through vertex cuts of an expanded version of the DAG itself. As a consequence of our main result, we provide an algorithm for enumerating all the minimal s-d cutsets in a temporal graph with delay O(m³), where m is the number of temporal arcs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • minimal cutset
  • temporal graph
  • minimal separator
  • directed acyclic graph


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Matthias Bentert, Anne-Sophie Himmel, Hendrik Molter, Marco Morik, Rolf Niedermeier, and René Saitenmacher. Listing all maximal k-plexes in temporal graphs. In ASONAM, pages 41-46, 2018. Google Scholar
  2. Kenneth A Berman. Vulnerability of scheduled networks and a generalization of menger’s theorem. Networks: An International Journal, 28(3):125-134, 1996. Google Scholar
  3. Anne Berry, Jean-Paul Bordat, and Olivier Cogis. Generating all the minimal separators of a graph. In WG, pages 167-172, 1999. Google Scholar
  4. Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. Algorithmica, 32(3):474-506, 2002. Google Scholar
  5. Binh-Minh Bui-Xuan, Afonso Ferreira, and Aubin Jarry. Computing shortest, fastest, and foremost journeys in dynamic networks. International Journal of Foundations of Computer Science, 14(2):267-285, 2003. Google Scholar
  6. Arnaud Casteigts, Paola Flocchini, Walter Quattrociocchi, and Nicola Santoro. Time-varying graphs and dynamic networks. International Journal of Parallel, Emergent and Distributed Systems, 27(5):387-408, 2012. Google Scholar
  7. S.K. Chaturvedi, Gaurav Khanna, and Sieteng Soh. Reliability evaluation of time evolving delay tolerant networks based on sum-of-disjoint products. Reliability Engineering & System Safety, 171:136-151, 2018. Google Scholar
  8. Qizhi Fang, Rudolf Fleischer, Jian Li, and Xiaoxun Sun. Algorithms for core stability, core largeness, exactness, and extendability of flow games. In COCOON, pages 439-447, 2007. Google Scholar
  9. Longxiang Gao, Shui Yu, Tom H. Luan, and Wanlei Zhou. Delay Tolerant Networks. Springer, 2015. Google Scholar
  10. Anne-Sophie Himmel, Hendrik Molter, Rolf Niedermeier, and Manuel Sorge. Adapting the Bron-Kerbosch algorithm for enumerating maximal cliques in temporal graphs. Social Network Analysis and Mining, 7(1):35, 2017. Google Scholar
  11. P. A. Jensen and M. Bellmore. An algorithm to determine the reliability of a complex system. IEEE Transactions on Reliability, R-18(4):169-174, 1969. Google Scholar
  12. Gaurav Khanna, Sanjay K. Chaturvedi, and Sieteng Soh. Two-terminal reliability analysis for time-evolving and predictable delay-tolerant networks. Recent Advances in Electrical & Electronic Engineering, 13(2):236-250, 2020. Google Scholar
  13. Gaurav Khanna and Sanjay Kumar Chaturvedi. A comprehensive survey on multi-hop wireless networks: Milestones, changing trends and concomitant challenges. Wireless Personal Communications, 101(2):677-722, 2018. Google Scholar
  14. Rohit Kumar and Toon Calders. 2SCENT: an efficient algorithm to enumerate all simple temporal cycles. Proceedings of the VLDB Endowment, 11(11):1441-1453, 2018. Google Scholar
  15. Matthieu Latapy, Tiphaine Viard, and Clémence Magnien. Stream graphs and link streams for the modeling of interactions over time. Social Network Analysis and Mining, 8(1):61:1-61:29, 2018. Google Scholar
  16. Q. Liang and E. Modiano. Survivability in time-varying networks. IEEE Transactions on Mobile Computing, 16(9):2668-2681, 2017. Google Scholar
  17. Andrea Marino. Analysis and Enumeration: Algorithms for Biological Graphs. Springer, 2015. Google Scholar
  18. Othon Michail. An introduction to temporal graphs: An algorithmic perspective. Internet Mathematics, 12(4):239-280, 2016. Google Scholar
  19. Petra Mutzel and Lutz Oettershagen. On the enumeration of bicriteria temporal paths. In TAMC, pages 518-535, 2019. Google Scholar
  20. Lhouari Nourine and Olivier Raynaud. A fast algorithm for building lattices. Information Processing Letters, 71(5-6):199-204, 1999. Google Scholar
  21. J. Scott Provan and Douglas R. Shier. A paradigm for listing (s, t)-cuts in graphs. Algorithmica, 15(4):351-372, 1996. Google Scholar
  22. Marvin Rausand. Reliability of Safety‐Critical Systems: Theory and Applications. John Wiley & Sons, Ltd, 2014. Google Scholar
  23. Hong Shen, Keqin Li, and Si-Qing Zheng. Separators are as simple as cutsets. In ASIAN, pages 347-358, 1999. Google Scholar
  24. Douglas R. Shier. Network Reliability and Algebraic Structures. Clarendon Press, 1991. Google Scholar
  25. S. Tsukiyama, I. Shirakawa, H. Ozaki, and H. Ariyoshi. An algorithm to enumerate all cutsets of a graph in linear time per cutset. Journal of the ACM, 27(4):619-632, 1980. Google Scholar
  26. Tiphaine Viard, Matthieu Latapy, and Clémence Magnien. Computing maximal cliques in link streams. Theoretical Computer Science, 609:245-252, 2016. Google Scholar
  27. Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The complexity of finding small separators in temporal graphs. J. Comput. Syst. Sci., 107:72-92, 2020. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail