Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, series-parallel digraphs, planar st-digraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straight-line, polyline, visibility), and update the drawing in a smooth way.

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.

A tension-free layout of a weighted graph G is an embedding of G in the plane such that the Euclidean distance between adjacent nodes is equal to the edge weight. Very few weighted graphs admit such a layout. However, any graph can be made into a tension-free graph by repeated application of an operation called vertex splitting, or by removing edges. In this paper we show that computing the minimum number of such operations that yield a tension-free graph is NP-hard. 1 Introduction A tension-free layout of a weighted graph G is an embedding of G in the plane such that the Euclidean distance between adjacent nodes is equal to the edge weight. Tension-free layouts of graphs play an important role in several visualization problems. For example, in the problem of visualization of email traffic [8, 3] we are required to draw a graph of email connections on the screen, with the Euclidean distance between two nodes being proportional to the amount of email traffic between the nodes. Many oth...