文章基本信息

标题：The Complexity of Finding Small Separators in Temporal Graphs
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作者：Philipp Zschoche ; Till Fluschnik ; Hendrik Molter 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2018
卷号：117
页码：1-17
DOI：10.4230/LIPIcs.MFCS.2018.45
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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.
关键词：(non-)strict temporal paths; temporal core; single-source shortest paths; node multiway cut; length-bounded cuts; parameterized complexity