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 ...

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 level graph G -- (V, E, q) is a directed acyclic graph with a mapping q: V - {1, 2,...,k), k _ 1, that partitions the vertex set V as V-- V10V20 ...V k, vj = q-l(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i, all v V i are drawn on the line li -- {(x, k - i) ] x ), the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to

Abstract. Graphs are widely used for information visualization purposes, since they provide a natural and intuitive representation of complex abstract structures. The automatic generation of drawings of graphs has applications a variety of fields such as software engineering, database systems, and graphical user interfaces. In this paper, we survey algorithmic techniques for graph drawing that support the expression and satisfaction of user-defined constraints. 1.

Let G be an n-node planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for 15 four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worst-case lower bound on the required width. Also, if G has no degree-three (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �

Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which produces planar drawings of clustered graphs in a convention known as orthogonal-grid rectangular cluster drawings. The drawing produced by the algorithm has constant number of bends on each edge and has O(n 2 ) area, which is as good as existing results for classical graph drawings. 1 Introduction Clustered graphs are graphs with recursive clustering structures over the vertices (see Fig. 1). This type of clustering structure appears in many systems. Examples include CASE tools [19], management information systems [10], and VLSI design tools [8]. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which ...

The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph drawing. Specific topics include bounds and tradeoffs for drawing properties, three-dimensional representations, methods for constraint satisfaction, and experimental studies. 1 Introduction In this paper, we survey selected research trends in graph drawing, and overview some recent results of the author and his collaborators. Graph drawing addresses the problem of constructing geometric representations of graphs, a key component of support tools for complex applications in science and engineering. Graph drawing is a young research field that has growth very rapidly in the last decade. One of its distinctive characteristics is to have furthered collaborative efforts between computer scien...

In this paper we consider the class of directed acyclic graphs (DAGs), and present the results of an experimental study on four drawing algorithms specifically developed for DAGs. Our study is conducted on two large test suites of DAGs and yields more than 30 charts comparing the performance of the drawing algorithms with respect to several quality measures, including area, crossings, bends, and aspect ratio. The algorithms exhibit various trade-offs with respect to the quality measures, and none of them clearly outperforms the others.

Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which produces planar drawings of clustered graphs in a convention known as orthogonal grid rectangular cluster drawings. We present an algorithm which produces such drawings with O(n 2 ) area and with at most 3 bends in each edge. This result is as good as existing results for classical planar graphs. Further, we show that the bend performance of our algorithm is optimal. (Extended Abstract) 1 Introduction Clustered graphs are graphs with recursive clustering structures over the vertices (see Figure 1). This type of clustering structure appears in many systems. Examples include CASE tools [40], management information systems [19], and VLSI design tools [15]. For graphical representation, the clust...