We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straight-line drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on the angular resolution of planar straight-line drawings, and show a continuous trade-off between the area and the angular resolution. We also give linear-time algorithms for constructing planar straight-line drawings with high angular resolution for various classes of graphs, such as series-parallel graphs, outerplanar graphs, and triangulations generated by nested triangles. Our results are obtained by new techniques that make extensive use of geometric constructions.

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 examines an infinite family of proximity drawings of graphs called open and closed fi-drawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the well-known Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit open fi-drawings for 0 fi ! fi ! 1 or closed fi-drawings for 0 fi ! fi 1 are given, as well as partial characterizations for other values of fi. For the intervals of fi in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed fi-drawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.

This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required

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

Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, gabriel graph or modified gabriel graph of a set of points in the plane. The characterizations give rise to linear-time algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of gabriel graphs settles the conjectures of Matula and Sokal [19].

We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any respect to the size of the vertices. Naturally, the vertices should be represented by uniformly small squares. In addition we might require that each face should be represented by a non-empty region. This would allow a labeling of the faces. We present an efficient algorithm which provably achieves the bend minimum following these constraints. Omitting the latter requirement we conjecture that the problem becomes NP-hard. For that case, we give advices for good approximations. We demonstrate the effectiveness of our approaches giving some interesting examples.

Motivated by rectangular visibility and graph drawing applications, we study the problem of characterizing classes of graphs that admit rectangle of influence drawings. We consider several classes of graphs and show, for each class, that testing whether a graph G has a rectangle of influence drawing can be done in O(n) time, where n is the number of vertices of G. If the test for G is affirmative, we show how to construct a rectangle of influence drawing of G. All the drawing algorithms can be implemented so that they (1) produce drawings with all vertices placed at intersection points of an integer grid of size O(n 2 ), (2) perform arithmetic operations on integers only, and (3) run in O(n) time, where n is the number of vertices of the input graph. 1 Introduction A proximity drawing of a graph is a straight-line drawing (vertices are represented by points and edges by straight-line segments) where the points representing adjacent vertices are deemed to be close according t...

Surface parameterization is a fundamental problem in computer graphics. Intuitively, we can think of it as the flattening of a surface to a valid planar configuration, i.e. one without foldovers or self-intersections. More formally, consider a surface that is homeomorphic to a disk. Then the goal is to find

We consider the problem of finding a planar straight-line embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable—indeed, solvable in linear time on a real RAM—for straight-line embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead straight-line embeddings of planar 3-connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.