A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

TL;DR

This paper introduces the cd-tree data structure to provide a new characterization of c-planarity, enabling efficient testing for certain clustered graphs and revealing connections to constrained planarity problems.

Contribution

It presents a novel data structure and characterization for c-planarity, leading to efficient algorithms for specific cases and new insights into the problem's structure.

Findings

Efficient c-planarity testing for clusters with limited connected components.

Efficient c-planarity testing when clusters have few outgoing edges.

Revealed connections between c-planarity and constrained planarity problems.

Abstract

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.