# DAGmaps and Dominance Relationships

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