The Complexity of Finding Small Separators in Temporal Graphs

TL;DR

This paper investigates the computational complexity of finding small separators in temporal graphs under different models and parameters, revealing NP-hardness results and identifying tractable cases based on graph planarity, number of time steps, and temporal core size.

Contribution

It provides a complexity dichotomy for separator problems in temporal graphs, introduces the concept of a temporal core, and analyzes fixed-parameter tractability for these problems.

Findings

NP-hardness for general cases and planar underlying graphs.

Polynomial-time solvability when the number of time steps is constant for strict paths.

Fixed-parameter tractability of the non-strict variant with respect to the temporal core size.

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.