Bipartite Temporal Graphs and the Parameterized Complexity of Multistage 2-Coloring

TL;DR

This paper studies the complexity of recognizing bipartite temporal graphs with minimal changes between layers, showing NP-hardness in simple cases and fixed-parameter tractability when limiting total changes.

Contribution

It introduces a new framework for bipartite temporal graphs based on minimal layer changes and analyzes the problem's complexity and parameterized tractability.

Findings

Recognition is NP-hard with only two layers or one change allowed.

Certain structural parameters transfer from static to temporal graphs.

Total change restriction leads to fixed-parameter tractability.

Abstract

We consider the algorithmic complexity of recognizing bipartite temporal graphs. Rather than defining these graphs solely by their underlying graph or individual layers, we define a bipartite temporal graph as one in which every layer can be 2-colored in a way that results in few changes between any two consecutive layers. This approach follows the framework of multistage problems that has received a growing amount of attention in recent years. We investigate the complexity of recognizing these graphs. We show that this problem is NP-hard even if there are only two layers or if only one change is allowed between consecutive layers. We consider the parameterized complexity of the problem with respect to several structural graph parameters, which we transfer from the static to the temporal setting in three different ways. Finally, we consider a version of the problem in which we only restrict the total number of changes throughout the lifetime of the graph. We show that this variant is fixed-parameter tractable with respect to the number of changes.