The Complexity of Finding Small Separators in Temporal Graphs

Authors Philipp Zschoche, Till Fluschnik, Hendrik Molter, Rolf Niedermeier

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Philipp Zschoche
  • Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany
Till Fluschnik
  • Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany
Hendrik Molter
  • Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany
Rolf Niedermeier
  • Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany

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Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The Complexity of Finding Small Separators in Temporal Graphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Temporal graphs are graphs with time-stamped edges. We study the problem of finding a small vertex set (the separator) with respect to two designated terminal vertices such that the removal of the set eliminates all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that pass through arbitrarily many edges per time step (non-strict) and paths that pass through at most one edge per time step (strict). Regarding the number of time steps of a temporal graph, we show a complexity dichotomy (NP-hardness versus polynomial-time solvability) for both problem variants. Moreover we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. We further show that, on temporal graphs with planar underlying graph, if additionally the number of time steps is constant, then the problem variant for strict paths is solvable in quasi-linear time. Finally, we introduce and motivate the notion of a temporal core (vertices whose incident edges change over time). We prove that the non-strict variant is fixed-parameter tractable when parameterized by the size of the temporal core, while the strict variant remains NP-complete, even for constant-size temporal cores.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Problems, reductions and completeness
  • (non-)strict temporal paths
  • temporal core
  • single-source shortest paths
  • node multiway cut
  • length-bounded cuts
  • parameterized complexity


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  1. Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows: Theory, Algorithms and Applications. Prentice Hall, 1993. Google Scholar
  2. Eleni C Akrida, Jurek Czyzowicz, Leszek Gąsieniec, Łukasz Kuszner, and Paul G Spirakis. Temporal flows in temporal networks. In Proceedings of the 10th International Conference on Algorithms and Complexity (CIAC '17), pages 43-54. Springer, 2017. Google Scholar
  3. Eleni C Akrida, Leszek Gąsieniec, George B Mertzios, and Paul G Spirakis. On temporally connected graphs of small cost. In Proceedings of the 13th International Workshop on Approximation and Online Algorithms (WAOA '15), pages 84-96. Springer, 2015. Google Scholar
  4. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, 1991. Google Scholar
  5. Kyriakos Axiotis and Dimitris Fotakis. On the size and the approximability of minimum temporally connected subgraphs. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP '16), pages 149:1-149:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  6. Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondřej Pangrác, Heiko Schilling, and Martin Skutella. Length-bounded cuts and flows. ACM Transactions on Algorithms, 7(1):4:1-4:27, 2010. Google Scholar
  7. Kenneth A Berman. Vulnerability of scheduled networks and a generalization of Menger’s Theorem. Networks, 28(3):125-134, 1996. Google Scholar
  8. Stefano Boccaletti, Ginestra Bianconi, Regino Criado, Charo I Del Genio, Jesús Gómez-Gardenes, Miguel Romance, Irene Sendina-Nadal, Zhen Wang, and Massimiliano Zanin. The structure and dynamics of multilayer networks. Physics Reports, 544(1):1-122, 2014. Google Scholar
  9. 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
  10. H.W Corley and David Y Sha. Most vital links and nodes in weighted networks. Operations Research Letters, 1(4):157-160, 1982. Google Scholar
  11. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-order Logic: a Language-theoretic Approach. Cambridge University Press, 2012. Google Scholar
  12. Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. On multiway cut parameterized above lower bounds. ACM Transactions on Computation Theory, 5(1):3:1-3:11, 2013. Google Scholar
  13. Reinhard Diestel. Graph Theory, 5th Edition, volume 173 of Graduate Texts in Mathematics. Springer, 2016. Google Scholar
  14. Thomas Erlebach, Michael Hoffmann, and Frank Kammer. On temporal graph exploration. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP '15), pages 444-455. Springer, 2015. Google Scholar
  15. Afonso Ferreira. Building a reference combinatorial model for MANETs. IEEE Network, 18(5):24-29, 2004. Google Scholar
  16. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer-Verlag, Berlin, 2006. Google Scholar
  17. Till Fluschnik, Danny Hermelin, André Nichterlein, and Rolf Niedermeier. Fractals for kernelization lower bounds. SIAM Journal on Discrete Mathematics, 32(1):656-681, 2018. Google Scholar
  18. Till Fluschnik, Hendrik Molter, Rolf Niedermeier, and Philipp Zschoche. Temporal graph classes: A view through temporal separators. arXiv preprint arXiv:1803.00882, 2018. To appear in Proceedings 44th International Workshop on Graph-Theoretic Concepts in Computer Science (WG '18). Google Scholar
  19. Lester R Ford and Delbert R Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8(3):399-404, 1956. Google Scholar
  20. Petr A. Golovach and Dimitrios M. Thilikos. Paths of bounded length and their cuts: Parameterized complexity and algorithms. Discrete Optimization, 8(1):72-86, 2011. Google Scholar
  21. Anne-Sophie Himmel. Algorithmic investigations into temporal paths. Masterthesis, TU Berlin, April 2018. URL:
  22. 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):35:1-35:16, 2017. Google Scholar
  23. Petter Holme. Modern temporal network theory: a colloquium. European Physical Journal B, 88(9):234, 2015. Google Scholar
  24. Petter Holme and Jari Saramäki. Temporal networks. Physics Reports, 519(3):97-125, 2012. Google Scholar
  25. David Kempe, Jon Kleinberg, and Amit Kumar. Connectivity and inference problems for temporal networks. Journal of Computer and System Sciences, 64(4):820-842, 2002. Google Scholar
  26. Matthieu Latapy, Tiphaine Viard, and Clémence Magnien. Stream graphs and link streams for the modeling of interactions over time. arXiv preprint arXiv:1710.04073, 2017. Google Scholar
  27. Qingkai Liang and Eytan Modiano. Survivability in time-varying networks. IEEE Transactions on Mobile Computing, 16(9):2668-2681, 2017. Google Scholar
  28. László Lovász, Víctor Neumann-Lara, and Michael Plummer. Mengerian theorems for paths of bounded length. Periodica Mathematica Hungarica, 9(4):269-276, 1978. Google Scholar
  29. Karl Menger. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae, 10(1):96-115, 1927. Google Scholar
  30. George B Mertzios, Othon Michail, Ioannis Chatzigiannakis, and Paul G Spirakis. Temporal network optimization subject to connectivity constraints. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP '13), pages 657-668. Springer, 2013. Google Scholar
  31. Othon Michail. An introduction to temporal graphs: An algorithmic perspective. Internet Mathematics, 12(4):239-280, 2016. Google Scholar
  32. Othon Michail and Paul G. Spirakis. Traveling salesman problems in temporal graphs. Theoretical Computer Science, 634:1-23, 2016. Google Scholar
  33. Feng Pan and Aaron Schild. Interdiction problems on planar graphs. Discrete Applied Mathematics, 198:215-231, 2016. Google Scholar
  34. Salvatore Scellato, Ilias Leontiadis, Cecilia Mascolo, Prithwish Basu, and Murtaza Zafer. Evaluating temporal robustness of mobile networks. IEEE Transactions on Mobile Computing, 12(1):105-117, 2013. Google Scholar
  35. Baruch Schieber, Amotz Bar-Noy, and Samir Khuller. The complexity of finding most vital arcs and nodes. Technical report, University of Maryland at College Park, College Park, MD, USA, 1995. Google Scholar
  36. Tiphaine Viard, Matthieu Latapy, and Clémence Magnien. Computing maximal cliques in link streams. Theoretical Computer Science, 609:245-252, 2016. Google Scholar
  37. Huanhuan Wu, James Cheng, Yiping Ke, Silu Huang, Yuzhen Huang, and Hejun Wu. Efficient algorithms for temporal path computation. IEEE Transactions on Knowledge and Data Engineering, 28(11):2927-2942, 2016. Google Scholar
  38. Philipp Zschoche. On finding separators in temporal graphs. Masterthesis, TU Berlin, August 2017. URL: