The Computational Complexity of Finding Separators in Temporal Graphs

Vertex separators, that is, vertex sets whose deletion disconnects two distinguished vertices ina graph, play a pivotal role in algorithmic graph theory. For instance, the concept of tree decom-positions of graphs is tightly connected to the separator concept. For many realistic models of thereal world, however, it is necessary to consider graphs whose edge set changes with time. Morespecifically, the edges are labeled with time stamps. In the literature, these graphs are referredto as temporal graphs, temporal networks, time-varying networks, edge-scheduled networks, etc.While there is an extensive literature on separators in “static” graphs, much less is known for thetemporal setting. Building on previous work (e.g., Kempe et al. [STOC ’00]), for the first timewe systematically investigate the (parameterized) complexity of finding separators in temporalgraphs. Doing so, we discover a rich landscape of computationally (fixed-parameter) tractableand intractable cases. In particular, we shed light on the so far seemingly overlooked fact thattwo frequently used models of temporal separation may lead to quite significant differences interms of computational complexity. More specifically, considering paths in temporal graphs onemay distinguish between strict paths (the time stamps along a path are strictly increasing) andnon-strict paths (the time stamps along a path are monotonically non-decreasing). We observethat the corresponding strict case of temporal separators leads to several computationally mucheasier to handle cases than the non-strict case does. ∗Supported by the Stiftung Begabtenförderung berufliche Bildung (SBB).†Supported by the DFG, project DAMM (NI 369/13). 1arXiv:1711.00963v3[cs.DS]8Dec2017