Results 1  10
of
39
A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... 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 ..."
Abstract

Cited by 627 (44 self)
 Add to MetaCart
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, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
Abstract

Cited by 78 (0 self)
 Add to MetaCart
We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
Planar Drawings and Angular Resolution: Algorithms and Bounds (Extended Abstract)
 IN PROC. 2ND ANNU. EUROPEAN SYMPOS. ALGORITHMS
, 1994
"... 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 straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on th ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
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 straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on the angular resolution of planar straightline drawings, and show a continuous tradeoff between the area and the angular resolution. We also give lineartime algorithms for constructing planar straightline drawings with high angular resolution for various classes of graphs, such as seriesparallel graphs, outerplanar graphs, and triangulations generated by nested triangles. Our results are obtained by new techniques that make extensive use of geometric constructions.
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
Abstract

Cited by 31 (3 self)
 Add to MetaCart
The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
Constrained Higher Order Delaunay Triangulations
 COMPUT. GEOM. THEORY APPL
, 2004
"... We extend the notion of higherorder Delaunay triangulations to constrained higherorder Delaunay triangulations and provide various results. We can determine the order k of a given triangulation in O(min(nk log n log k, n n)) time. We show that the completion of a set of useful orderk Dela ..."
Abstract

Cited by 27 (7 self)
 Add to MetaCart
We extend the notion of higherorder Delaunay triangulations to constrained higherorder Delaunay triangulations and provide various results. We can determine the order k of a given triangulation in O(min(nk log n log k, n n)) time. We show that the completion of a set of useful orderk Delaunay edges may have order 2k 2, which is worstcase optimal. We give an algorithm for the lowestorder completion for a set of useful orderk Delaunay edges when k 3. For higher orders the problem is open.
Arc triangulations
 PROC. 26TH EUR. WORKSH. COMP. GEOMETRY (EUROCG’10)
, 2010
"... The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alter ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
(Show Context)
The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... 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 pro ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
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 lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree 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 lefttoright 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 ...
The Strength of Weak Proximity
, 1996
"... This paper initiates the study of weak proximitydrawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximitydrawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other poi ..."
Abstract

Cited by 19 (11 self)
 Add to MetaCart
This paper initiates the study of weak proximitydrawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximitydrawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment(p# q) is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximitydrawing in which such segments must appear in the drawing. Most previously studied proximity regions are associated with a parameter fi,0 fi 1.For fixed fi,weak fidrawability is at least as expressive as strong fidrawability,as a strong fidrawing is also a weak one. Wegive examples of graph families and fi values where the two notions coincide, and a situation in which it is NPhard to determine weak fidrawability. On the other hand, wegive situations where weak proximity significantly increases the expressivepower of fidrawability: weshowthatevery graph has, for all sufficiently small fi,aweak fiproximitydrawing that is computable in linear time, and we show that every tree has, for every fi less than 2, a weak fidrawing that is computable in linear time.
A Framework for Drawing Planar Graphs with Curves and Polylines
 J. Algorithms
, 1998
"... We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
(Show Context)
We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any nvertex planar graph in an O(n) O(n) grid using polylines that have at most two bends per edge and asymptoticallyoptimal worstcase angular resolution. More significantly, we show how to adapt this algorithm to draw any nvertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. Al...
Improving graph drawing readability by incorporating readability metrics: A software tool for network analysts
, 2009
"... Designing graph drawings that effectively communicate the underlying network is challenging as for every network there are many potential unintelligible or even misleading drawings. Automated graph layout algorithms have helped, but frequently generate ineffective drawings. In order to build awarene ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
(Show Context)
Designing graph drawings that effectively communicate the underlying network is challenging as for every network there are many potential unintelligible or even misleading drawings. Automated graph layout algorithms have helped, but frequently generate ineffective drawings. In order to build awareness of effective graph drawing strategies, we detail readability metrics on a [0,1] continuous scale for node occlusion, edge crossing, edge crossing angle, and edge tunneling and summarize many more. Additionally, we define new node & edge readability metrics to provide more localized identification of where improvement is needed. These are implemented in SocialAction, a tool for social network analysis, in order to direct users towards poor areas of the drawing and provide realtime readability metric feedback as users manipulate it. These contributions are aimed at heightening the awareness of network analysts that the images they share or publish could be of higher quality, so that readers could extract relevant information.