Simultaneous Embedding and Level Planarity

Abstract

Graphs are a common model for representing information consisting of a set of objects or entities and a set of connections or relations between them. Graph Drawing is concerned with the automatic visualization of graphs in order to make the information useful. That is, a good drawing should be helpful in the application domain where it is used by capturing the relationships in the underlying data. We consider two important problems in automated graph drawing: simultaneous embedding and level planarity. Simultaneous embedding is the problem of drawing multiple graphs while maintaining the readability of each graph independently and preserving the mental map when going from one graph to another. In this case, each graph has the same vertex set (same entities) but different edge sets (different relationships). Level planarity arises in the layout of graphs that contain hierarchical relationships. When drawing graphs in the plane, this translates to a restricted form of planarity where the vertical order of the entities is pre-determined. We consider the computational complexity of the simultaneous embedding problem. In particular, we show that in its generality the simultaneous embedding problem is NP-hard if the edges are drawn as straight-lines. We present algorithms for drawing graphs on predetermined levels, which allow the simultaneous embedding of restricted types of graphs, such as outerplanar graphs, trees and paths. Finally, our practical contribution is a tool that implements known and novel algorithms related to simultaneous embedding and level planarity and can be used both as a visualization software and as an aid to study theoretical problems.