DAGmaps and Dominance Relationships
In  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. . 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.