Clustered Planarity = Flat Clustered Planarity

TL;DR

This paper proves that the complexity of determining flat clustered planarity is the same as the general problem, and shows polynomial-time equivalence to a simplified version where each cluster is independent.

Contribution

It establishes that flat clustered planarity has the same computational complexity as the general clustered planarity problem and is polynomial-time equivalent to independent flat clustered planarity.

Findings

Flat clustered planarity is as complex as the general problem.

Flat clustered planarity is polynomial-time equivalent to independent flat clustered planarity.

The results have implications for understanding the complexity of clustered graph drawing.

Abstract

The complexity of deciding whether a clustered graph admits a clustered planar drawing is a long-standing open problem in the graph drawing research area. Several research efforts focus on a restricted version of this problem where the hierarchy of the clusters is "flat", i.e., no cluster different from the root contains other clusters. We prove that this restricted problem, that we call Flat Clustered Planarity, retains the same complexity of the general Clustered Planarity problem, where the clusters are allowed to form arbitrary hierarchies. We strengthen this result by showing that Flat Clustered Planarity is polynomial-time equivalent to Independent Flat Clustered Planarity, where each cluster induces an independent set. We discuss the consequences of these results.