On Efficiently Finding Small Separators in Temporal Graphs
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Vertex separators, that is, vertex sets whose deletion disconnects two distinguished vertices in a graph, play a pivotal role in algorithmic graph theory. For many realistic models of the real world it is necessary to consider graphs whose edge set changes with time. More specifically, the edges are labeled with time stamps. In the literature, these graphs are referred to 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 the temporal setting. Building on previous work, we study the problem of finding a small vertex set (the separator) in a temporal graph with two designated terminal vertices such that the removal of the set breaks all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that contain arbitrarily many hops per time step (non-strict) and paths that contain at most one hop per time step (strict). We settle the hardness dichotomy (NP-hardness versus polynomial-time solvability) of both problem variants regarding the number of time steps of a temporal graph. Moreover we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. We show that on temporal graphs whose underlying graph is planar, if additionally the number of time steps is constant then the problem variant for strict paths is solvable in quasi linear time. Finally, for general temporal graphs we introduce the notion of a temporal core (vertices whose incident edges change over time). We prove that on temporal graphs with constant-sized temporal core, the non-strict variant is solvable in polynomial time, where the degree of the polynomial is independent of the size of the temporal core, while the strict variant remains NP-complete.
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