Relaxing the Constraints of Clustered Planarity

TL;DR

This paper explores relaxed conditions for clustered graph planarity by allowing certain crossings, providing bounds, existence results, and a polynomial-time test for specific cases, advancing understanding of c-planarity complexities.

Contribution

It introduces a relaxed framework for c-planarity, analyzes crossing relationships, and offers a polynomial-time algorithm for testing region-region crossings in biconnected graphs.

Findings

Drawings with only edge-edge or edge-region crossings always exist.

Drawings with only region-region crossings may not exist.

Polynomial-time algorithm for testing region-region crossings in biconnected graphs.

Abstract

In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge-edge, edge-region, or region-region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge-edge, edge-region, and region-region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge-edge or with only edge-region crossings always exist, while drawings with only region-region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region-region crossings exist for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs.