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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