Results 1  10
of
69
A Better Heuristic for Orthogonal Graph Drawings
 COMPUT. GEOM. THEORY APPL
, 1998
"... An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is ben ..."
Abstract

Cited by 65 (6 self)
 Add to MetaCart
An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is bent at most twice. In particular for nonplanar and nonbiconnected planar graphs, this is a big improvement. The algorithm is very simple, easy to implement, and it handles both planar and nonplanar graphs at the same time.
Convex Grid Drawings of 3Connected Planar Graphs
, 1994
"... We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane gr ..."
Abstract

Cited by 37 (7 self)
 Add to MetaCart
We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane graph G (with n 3), finds such a straightline convex embedding of G into a (n \Gamma 2) \Theta (n \Gamma 2) grid. 1 Introduction In this paper we consider the problem of aesthetic drawing of plane graphs, that is, planar graphs that are already embedded in the plane. What is exactly an aesthetic drawing is not precisely defined and, depending on the application, different criteria have been used. In this paper we concentrate on the two following criteria: (a) edges should be represented by straightline segments, and (b) faces should be drawn as convex polygons. F'ary [6], Stein [14] and Wagner [18] showed, independently, that each planar graph can be drawn in the plane in such a way that ...
A Lineartime Algorithm for Drawing a Planar Graph on a Grid
 Information Processing Letters
, 1989
"... We present a lineartime algorithm that, given an nvertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straightline segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid i ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
We present a lineartime algorithm that, given an nvertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straightline segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid in the plane in such a way that the edges are straight, nonintersecting line segments. The existence of such straightline embeddings for planar graphs was independently discovered by F'ary [Fa48], Stein [St51], and Wagner [Wa36]; this result also follows from Steinitz's theorem on convex polytopes in three dimensions [SR34]. The first algorithms for constructing straightline embeddings [Tu63, CYN84, CON85] required highprecision arithmetic, and the resulting drawings were not very aesthetic, since they tend to produce uneven distributions of vertices over the drawing area. Rosenstiehl and Tarjan [RT86] noticed that it would be convenient to be able to map veritices of a planar graph into a...
Convex drawings of Planar Graphs and the Order Dimension of 3Polytopes
 ORDER
, 2000
"... We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. ..."
Abstract

Cited by 32 (12 self)
 Add to MetaCart
We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.
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 25 (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.
On Finding the Rectangular Duals of Planar Triangular Graphs
 SIAM J. Comput
, 1993
"... We present a new linear time algorithm for finding rectangular duals of planar triangular graphs. The algorithm is conceptually simpler than the previous known algorithm. The coordinates of the rectangular dual constructed by our algorithm are integers and have pure combinatorial meaning. This allow ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
We present a new linear time algorithm for finding rectangular duals of planar triangular graphs. The algorithm is conceptually simpler than the previous known algorithm. The coordinates of the rectangular dual constructed by our algorithm are integers and have pure combinatorial meaning. This allows us to discuss the heuristics for minimizing the size of the rectangular duals. Key words: Algorithm, Planar graph, Rectangular dual. AMS(MOS) subject classifications: 05C05, 05C38, 68Q25, 68R10. 1. Introduction Let R be a rectangle. A rectangular subdivision system of R is a partition of R into a set \Phi = fR 1 ; R 2 ; : : : ; R n g of nonintersecting smaller rectangles such that no four rectangles in \Phi meet at the same point. A rectangular dual of a graph G = (V; E) is a rectangular subdivision system \Phi and a onetoone correspondence f : V ! \Phi such that two vertices u and v are adjacent in G if and only if their corresponding rectangles f(u) and f(v) share a common boundar...
Level Planar Embedding in Linear Time
, 1999
"... A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i, all v V i are drawn on the line li  {(x, k  i) ] x ), the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to
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 19 (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 ...
An Experimental Comparison of Three Graph Drawing Algorithms (Extended Abstract)
, 1995
"... In this paper we present an extensive experimental study... ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
In this paper we present an extensive experimental study...