Abstract
Temporal graphs have time-stamped edges. Building on previous work, we study the problem of finding a small vertex set (the separator) whose removal destroys all temporal paths between two designated terminal vertices. Herein, we consider two models of temporal paths: those that pass through arbitrarily many edges per time step (non-strict) and those 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-completeness 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. Finally, we introduce the notion of a temporal core (vertices whose incident edges change over time) and prove that the non-strict variant is fixed-parameter tractable when parameterized by the temporal core size, while the strict variant remains NP-complete, even for constant-size temporal cores.
Highlights
In complex network analysis, it is nowadays very common to have access to and process graph data where the interactions among the vertices are time-stamped
We introduce the notion of temporal cores in temporal graphs
We prove that Temporal (s, z)-Separation is fixed-parameter tractable (FPT) when parameterized by the size of the temporal core, while Strict Temporal (s, z)-Separation remains NP-complete even if the temporal core is empty
Summary
It is nowadays very common to have access to and process graph data where the interactions among the vertices are time-stamped. To show the polynomial-time solvability of Strict Temporal (s, z)-Separation for τ ≤ 4, we prove that a classic separator result of Lovász et al [28] translates to the strict temporal setting. Berman [7] proved that the vertex-variant of an analogue to Menger’s Theorem for temporal graphs, asking for the maximum number of (strict) temporal paths instead, does not hold. Mertzios et al [30] proved another analogue of Menger’s Theorem: the maximum number of strict temporal (s, z)-path which never leave the same vertex at the same time equals the minimum number of node departure times needed to separate s from z, where a node departure time (v, t) is the vertex v at time point t. We recently studied the computational complexity of (non-strict) temporal separation on several other restricted temporal graphs [18]
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