Directed Graphs and Rectangular Layouts (Extended abstract) Adam L. Buchsbaum, Emden R. Gansner, and Suresh Venkatasubramanian AT&T Labs – Research Florham Park, NJ 07932 {alb,erg,suresh}@research.att.com Abstract. This paper deals with the problem, arising in practice, of drawing a di- rected graph as a collection of disjoint, isothetic rectangles, where the rectangles of the nodes of each edge must touch and where the placement of the rectangles respects the ordering of the edges. It provides characterizations for those graphs having the special type of rectangular layout known as a rectangular dual. It then characterizes the st-graphs having rectangular layouts in terms of the existence of certain planar embeddings and the non-existence of a particular subgraph. 1 Introduction We consider the problem of drawing a directed graph using disjoint, isothetic rectangles for the nodes, such that (u, v) is an edge in the graph if and only if the corresponding rectangles R u and R v touch, with R u above or to the left of R v . Figure 1 provides an example, showing a directed tree drawn both using the traditional, straight-line drawing and as a collection of rectangles. The problem arose in the context of providing a graphical user interface to a re- lational database support system which allows users to model and administer their database. Users can specify their own schemas using entity-relationship models [17]. It is assumed that there are no cycles among the relationships in the schema. The system then displays the database entities as rectangular buttons. Clicking on a button provides information related to the corresponding entity type, such as detailed descriptions of the entity attributes or information about specific records. Experience indicates the benefit of juxtaposing buttons that correspond to related entities. In addition, if the relationship is one-to-many, the button corresponding to the “many” entity is positioned below or to the right of the “one” entity, to emphasize this sense of directionality. There are no additional constraints. In particular, even if two entities are not related, their rectan- gles may abut. Viewing the database schema as the obvious directed tree, with entities as vertices and relations as edges, and allowing undirected cycles, leads to the graph drawing problem stated above. 1 This generalizes the concept of rectangular layouts, in which an undirected graph G is drawn using disjoint rectangles for nodes, and two nodes are adjacent in G if and only if the two corresponding rectangles are adjacent. Without directionality, this [10,11, 1 The implied additional restriction that rectangles can only touch if there is an edge is unimpor- tant (cf. Section 2).