A New Temporal Interpretation of Cluster Editing

TL;DR

This paper extends the NP-complete Cluster Editing problem to temporal graphs, providing polynomial algorithms for special cases and a complete characterization with forbidden configurations, along with an FPT algorithm.

Contribution

It introduces a temporal interpretation of Cluster Editing, offers polynomial algorithms for restricted cases, and characterizes forbidden configurations for the temporal setting.

Findings

NP-completeness persists in temporal graphs on paths

Complete characterization involves forbidden configurations on at most five vertices

Provides an FPT algorithm parameterized by modifications and lifetime

Abstract

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.