GD 2009: Graph Drawing pp 424-425

# DAGmaps and Dominance Relationships

• Vassilis Tsiaras
• Ioannis G. Tollis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

## Abstract

In [2] we use the term DAGmap to describe space filling visualizations of DAGs according to constraints that generalize treemaps and we show that deciding whether or not a DAG admits a DAGmap drawing is NP-complete. Let G = (V,E) be a DAG with a single source s. A component st-graph G u,v of G is a subgraph of G with a single source u and a single sink v that contains at least two edges and that is connected with the rest of G through vertex u and/or vertex v. A vertex w dominates a vertex v if every path from s to v passes through w. The dominance relation in G can be represented in compact form as a tree T, called the dominator tree of G, in which the dominators of a vertex v are its ancestors. Vertex w is the immediate dominator of v if w is the parent of v in T. A simple and fast algorithm to compute T has been proposed by Cooper et al. [1]. The post − dominators of G are defined as the dominators in the graph obtained from G by reversing all directed edges and assuming that all vertices are reachable from a (possibly artificial) vertex t. Using the definition of DAGmaps, it is easy to prove that in a DAGmap of G the rectangle of a vertex u includes the rectangles of all vertices that are dominated (resp. post-dominated) by u. Therefore when vertex u dominates vertex v and vertex v post-dominates vertex u then the rectangles R u and R v of u and v coincide. Based on this observation, we propose a heuristic algorithm that transforms a DAG G into a DAG G′ that admits a DAGmap. When G contains component st-graphs then our algorithm performs significantly fewer duplications than the transformation of G into a tree.

## References

1. 1.
Cooper, K.D., Harvey, T.J., Kennedy, K.: A simple, fast dominance algorithm (2001), http://www.cs.rice.edu/~keith/EMBED/dom.pdf
2. 2.
Tsiaras, V., Triantafilou, S., Tollis, I.G.: DAGmaps: Space Filling Visualization of Directed Acyclic Graphs. JGAA 13(3), 319–347 (2009)

© Springer-Verlag Berlin Heidelberg 2010

## Authors and Affiliations

• Vassilis Tsiaras
• 1
• Ioannis G. Tollis
• 1
1. 1.Institute of Computer ScienceFoundation for Research and Technology-Hellas