Temporal Graph Classes: A View Through Temporal Separators

TL;DR

This paper explores the computational complexity of separating vertices in temporal graphs, identifying conditions under which the problem is tractable or NP-hard, with a focus on structural and temporal restrictions.

Contribution

It introduces new insights into the complexity boundaries of temporal (s, z)-Separation, including polynomial cases for bounded treewidth and other temporal restrictions.

Findings

Polynomial-time solvability for bounded treewidth graphs

Identification of complexity borders based on temporal evolution

Systematic classification of tractable and intractable cases

Abstract

We investigate the computational complexity of separating two distinct vertices s and z by vertex deletion in a temporal graph. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s, z)-Separation problem is NP-hard, it is natural to investigate whether relevant special cases exist that are computationally tractable. To this end, we study restrictions of the underlying (static) graph---there we observe polynomial-time solvability in the case of bounded treewidth---as well as restrictions concerning the "temporal evolution" along the time steps. Systematically studying partially novel concepts in this direction, we identify sharp borders between tractable and intractable cases.