Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.

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.

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.

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

) Isabel F. Cruz 1 and Ashim Garg 2 1 Department of Electrical Engineering and Computer Science Tufts University Medford, MA 02155, USA 2 Department of Computer Science Brown University Providence, RI 02912--1910, USA Abstract. Constraint-based graph drawing systems provide expressive power and flexibility. Previously proposed approaches make use of general constraint solvers, which are inefficient, and of textual specification of constraints, which can be long and difficult to understand. In this paper we propose the use of a constraint-based visual language for constructing planar drawings of trees, series-parallel graphs, and acyclic digraphs in linear time. A graph drawing system based on our approach can therefore provide the power of constraint-based graph drawing, the simplicity of visual specifications, and the computational efficiency that is typical of the algorithmic-based approaches. 1 Introduction It is common practice to explain the layout of a graph using pictu...

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...

Abstract. We revisit monotone planar circuits MPCVP, with special attention to circuits with cylindrical embeddings. MPCVP is known to be in NC 3 in general, and in LogDCFL for the special case of upward stratified circuits. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that monotone circuits with embeddings that are stratified cylindrical, cylindrical, planar one-input-face and focused can be evaluated in LogDCFL, AC 1 (LogDCFL), LogCFL and AC 1 (LogDCFL) respectively. We note that the NC 3 algorithm for general MPCVP is in AC 1 (LogCFL) =SAC 2.Finally, we show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. 1

An upward embedding of an embedded planar graph states, for each vertex v, which edges are incident to v \above" or \below" and, in turn, induces an upward orientation of the edges. In this paper we characterize the set of all upward embeddings and orientations of a plane graph by using a simple ow model. We take advantage of such a ow model to compute upward orientations with the minimum number of sources and sinks of 1-connected graphs. Our theoretical results allow us to easily compute visibility representations of 1-connected graphs while having a certain control over the width and the height of the computed drawings, and to deal with partial assignments of the upward embeddings \underlying" the visibility representations. 2 1

How may a robot arm be moved to pick up a particular object from a crowded shelf without unwanted collisions? How may a cluster of figures on a computer screen be shifted about to clear the screen without altering their integrity and without collisions? These are instances of the problem known in computational geometry as the

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...