Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each plane graph can be drawn in such a way in a (n \Gamma 2) \Theta (n \Gamma 2) grid (for n 3), and that no grid smaller than (2n=3 \Gamma 1) \Theta (2n=3 \Gamma 1) can be used for this purpose, if n is a multiple of 3. In fact, for all n 3, each dimension of the resulting grid needs to be at least b2(n \Gamma 1)=3c, even if the other one is allowed to be unbounded. In this paper we show that this bound is tight by presenting a grid drawing algorithm that produces drawings of width b2(n \Gamma 1)=3c. The height of the produced drawings is bounded by 4b2(n \Gamma 1)=3c \Gamma 1. Our algorithm runs in linear time and is easy to implement. 1 Introduction The problem of automatic graph drawing ha...

. We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n \Gamma 5) \Theta ( 3 2 n \Gamma 7 2 ) grid with at most 5n \Gamma 15 bends and minimum angle ? 2 d . In the constructed drawings, every edge has at most three bends and length O(n). To our best knowledge, this algorithm achieves the best simultaneous bounds concerning the grid size, angular resolution, and number of bends for planar grid drawings of high-degree planar graphs. Besides the nice theoretical features, the practical drawings are aesthetically very pleasing. An implementation of our algorithm is available with the AGD-Library (Algorithms for Graph Drawing) [2, 1]. Our algorithm is based on ideas by Kant for polyline grid drawings for triconnected plane graphs [23]. In particular, our algorithm significantly improves upon his bounds on the angular resolution and the grid size for non-triconnected plane graphs....

Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)-area planar upward grid drawing that preserves the left-to-right ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...

This paper appears as Technical Report 95-07, Department of Computer Science, University of Newcastle, Newcastle NSW 2308 Australia.

The problem of drawing graphs nicely contains several computationally intractable subproblems. Hence, it is natural to apply genetic algorithms to graph drawing. This paper introduces a genetic algorithm (TimGA) which nicely draws undirected graphs of moderate size. The aesthetic criteria used are the number of edge crossings, even distribution of nodes, and edge length deviation. Although TimGA usually works well, there are some unsolved problems related to the genetic crossover operation of graphs. Namely, our tests indicate that TimGA's search is mainly guided by the mutation operations.

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...

In this paper we describe a general technique for establishing NP-hardness of graph representations. This technique is a generalization of the tool called the logic engine. We show that it is possible to extend it to a wobbly logic engine, which provides a proof method of NP-hardness for a variety of graph representations for which the set of feasible representations does not have to be discrete. This includes representations by visibility and intersection. In particular, we give a first proof that it is NP-hard to decide whether a graph has a nondegenerate z-axis parallel visibility representation (ZPR) by unit squares.