A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment, and no two edges cross. Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures. In this paper we show that upward planarity testing and rectilinear planarity testing are NP-complete problems. We also show that it is NP-hard to approximate the minimum number of bends in a planar orthogonal drawing of an n-vertex graph with an O(n 1\Gammaffl ) error, for any ffl ? 0.

Layout refers to the process of determining the sizes and positions of the visual objects that are part of an information presentation. Automated layout refers to the use of a computer program to automate either all or part of the layout process. This field of research lies at the crossroads between artificial intelligence and human computer interaction. Automated layout of presentations is becoming increasingly important as the amount of data that we need to present rapidly overtakes our ability to present it manually. We survey and analyze the techniques used by research systems that have automated layout components and suggest possible areas of future work.

Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of single-source digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a single-source digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)-time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.

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...

INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computer-aided-design. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and human-computer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin

A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a non-algorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given single-source acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...

Graph drawing algorithms usually attempt to display the characteristic properties of the input graphs. In this paper we consider the class of planar bipartite graphs and try to achieve planar drawings such that the bipartiteness property is cleary shown. To this aim, we develop several models, give efficient algorithms to find a corresponding drawing if possible or prove the hardness of the problem. 1 Introduction Graph drawing is a more and more developing method to visualize data and their relations. The main goal is to draw the graph in such a way that certain properties are clearly displayed: Planar graphs should be drawn planar [6], symmetries should be displayed [3, 11], if the graph is directed and acyclic then it should be drawn 'upward' [12], cliques should be easily recognized. There are many more properties developed in graph theory and graph algorithms that are worth to be displayed [1]. Another important property is the bipartiteness. A graph G = (V; E) with a set V of v...

author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NP-complete. Parameterized complexity is a technique used to find efficient algorithms for hard

Graph Drawing (also known as Graph Visualization) tackles the problem of representing graphs on a visual medium such as computer screen, printer etc. Many applications such as software engineering, data base design, project planning, VLSI design, multimedia etc., have data structures that can be represented as graphs. With the ever increasing complexity of these and new applications, and availability of hardware supporting visualization, the area of graph drawing is increasingly getting more attention from both practitioners and researchers. In a typical drawing of a graph, the vertices are represented as symbols such as circles, dots or boxes, etc., and the edges are drawn as continuous curves joining their end points. Often, the edges are simply drawn as (straight- or poly-) lines joining their end points (and hence the title of this thesis), followed by an optional transformation into smooth curves. The goal of research in graph drawing is to develop techniques for constructing good...

The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if its blocks admit upward planar drawings with certain properties. We also show how to test if a block of a digraph admits an upward planar drawing with the aforementioned properties.