On Finding Separators in Temporal Split and Permutation Graphs

TL;DR

This paper investigates the computational complexity of the NP-hard temporal (s,z)-separation problem on temporal graphs with layers restricted to split or permutation graphs, identifying both hardness and fixed-parameter tractability results.

Contribution

It provides the first complexity analysis of the temporal (s,z)-separation problem on layered graphs restricted to perfect graph classes, revealing NP-hardness and fixed-parameter tractability.

Findings

NP-hardness on temporal split and permutation graphs

Fixed-parameter tractability based on change over time

Identification of tractable cases within perfect graph classes

Abstract

Removing all connections between two vertices s and z in a graph by removing a minimum number of vertices is a fundamental problem in algorithmic graph theory. This (s,z)-separation problem is well-known to be polynomial solvable and serves as an important primitive in many applications related to network connectivity. We study the NP-hard temporal (s,z)-separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle this problem by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general the problem remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of "change over time".