Clustered graphs are graphs with recursive clustering structures over the vertices. This type of structure appears in many systems. Examples include CASE tools, management information systems, VLSI design tools, and reverse engineering systems. Existing layout algorithms represent the clustering structure as recursively nested regions in the plane. However, as the structure becomes more and more complex, two dimensional plane representations tend to be insufficient. In this paper, firstly, we describe some two dimensional plane drawing algorithms for clustered graphs; then we show how to extend two dimensional plane drawings to three dimensional multilevel drawings. We consider two conventions: straight-line convex drawings and orthogonal rectangular drawings; and we show some examples. 1 Introduction Graph drawing algorithms are widely used in graphical user interfaces of software systems. As the amount of information that we want to visualize becomes larger, we need more structure ...

An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is bent at most twice. In particular for non-planar and non-biconnected planar graphs, this is a big improvement. The algorithm is very simple, easy to implement, and it handles both planar and non-planar graphs at the same time.

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straight-line representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR-85-11713, CCR-86-05353, and CCR-88-14977, and by DARPA contract N00039-84-C-0165.

We give a detailed description of the embedding phase of the Hopcroft and Tarjan planarity testing algorithm. The embedding phase runs in linear time. An implementation based on this paper can be found in [MMN93].

vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

In this paper, we present algorithms to produce orthogonal drawings of arbitrary graphs. As opposed to most known algorithms, we do not restrict ourselves to graphs with maximum degree 4. The best previous result gave an (m \Gamma 1) \Theta ( m 2 + 1)-grid for graphs with n nodes and m edges. We present algorithms for two scenarios. In the static scenario, the graph is given completely in advance. We produce a drawing on a grid of size at most m+n 2 \Theta m+n 2 , or on a larger grid where the aspect ratio of the nodes is bounded. Furthermore, we give upper and lower bounds for drawings of the complete graph Kn in the underlying model. In the incremental scenario, the graph is given one node at a time, and the placement of previous nodes can not be changed for later nodes. We then come close to the bounds achieved in the static case and get at most an ( m 2 + n) \Theta ( 2 3 m+ n)-grid. In both algorithms, every edge gets at most one bend, thus, the total number of bends ...

In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics workstation as graph drawings. The usefulness of the relational model depends on whether the graph drawings effectively convey the relational information to the users. This thesis is concerned with finding good drawings of graphs. As the amount of information that we want to visualize becomes larger and the relations become more complex, the classical graph model tends to be inadequate. Many extended models use a node hierarchy to help cope with the complexity. This thesis introduces a new graph model called the clustered graph. The central theme of the thesis is an investigation of efficient algorithms to produce good drawings for clustered graphs. Although the criteria for judging the qua...

A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but all are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the complexities of previous approaches (including the need to first st-number the vertices). Our new algorithm easily permits the extraction of a planar obstruction (a subgraph homeomorphic to K3;3 or K5) in O(n) time if the graph is not planar. Our algorithm is similar to the algorithm of Booth and Lueker which uses a data structure called a PQ-tree. The Pnodes in a PQ-tree represent parts of the partially embedded graph that can be permuted, and the Q-nodes represent parts that can be flipped. We avoid the use of P-nodes by not connecting pieces together until they become biconnected. We avoid Q nodes by using a data structure which allows biconnected components to be flipped in O(1) time. 1 In...

A box is a restricted portion of the three-dimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A box-drawing of a graph is a straight-line crossing-free drawing where vertices are located at integer grid points along the four lines. It is known that some planar graphs with tri-connected components do not admit a box-drawing. This paper shows that even structurally simpler planar graphs, namely series-parallel graphs, are not boxdrawable in general. On the positive side, it is proved that every series-parallel graph whose vertices have maximum degree at most three is box-drawable. A drawing algorithm is presented that computes a box drawing of a 3-planar series-parallel graph in optimal time and with optimal volume.