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Relaxing the Constraints of Clustered Planarity
- Publication Year :
- 2012
-
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, this 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 exists 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.
- Subjects :
- Computational Geometry (cs.CG)
FOS: Computer and information sciences
Control and Optimization
Discrete Mathematics (cs.DM)
Biconnected graph
law.invention
Combinatorics
Planar graph
law
Outerplanar graph
Line graph
NP-hardness
Graph property
Complement graph
Mathematics
Discrete mathematics
Book embedding
Computer Science Applications
Graph drawing
Computational Mathematics
Computational Theory and Mathematics
Computer Science - Computational Geometry
Geometry and Topology
Crossing number (graph theory)
Null graph
Clustered planarity
Computer Science - Discrete Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....246b9a0b7c1b70d715bea19c3eb783e3