The complexity of finding small separators in temporal graphs

Philipp Zschoche, Till Fluschnik, Hendrik Molter, Rolf Niedermeier

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

8 Scopus citations

Abstract

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.

Original languageEnglish
Title of host publication43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018
EditorsIgor Potapov, James Worrell, Paul Spirakis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770866
DOIs
StatePublished - 1 Aug 2018
Externally publishedYes
Event43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 - Liverpool, United Kingdom
Duration: 27 Aug 201831 Aug 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume117
ISSN (Print)1868-8969

Conference

Conference43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018
Country/TerritoryUnited Kingdom
CityLiverpool
Period27/08/1831/08/18

Keywords

  • (non-)strict temporal paths
  • Length-bounded cuts
  • Node multiway cut
  • Parameterized complexity
  • Single-source shortest paths
  • Temporal core

ASJC Scopus subject areas

  • Software

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