Results 1  10
of
20
Planar embeddability of the vertices of a graph using a fixed point set is NPhard
, 2003
"... ..."
(Show Context)
Planar embeddings of graphs with specified edge lengths
, 2007
"... We consider the problem of finding a planar straightline 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 tha ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
We consider the problem of finding a planar straightline 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 NPhard. In contrast, we show that the problem is tractable—indeed, solvable in linear time on a real RAM—for straightline embeddings of planar 3connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NPhard if we consider instead straightline embeddings of planar 3connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.
On the Queue Number of Planar Graphs
, 2010
"... We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1
Accelerated bend minimization
, 2012
"... We present an O(n 3/2) algorithm for minimizing the number of bends in an orthogonal drawing of a plane graph. It has been posed as a long standing open problem at Graph Drawing 2003, whether the bound of O(n 7/4 √ log n) shown by Garg and Tamassia in 1996 could be improved. To answer this question, ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We present an O(n 3/2) algorithm for minimizing the number of bends in an orthogonal drawing of a plane graph. It has been posed as a long standing open problem at Graph Drawing 2003, whether the bound of O(n 7/4 √ log n) shown by Garg and Tamassia in 1996 could be improved. To answer this question, we show how to solve the uncapacitated mincost flow problem on a planar bidirected graph with bounded costs and face sizes in O(n 3/2) time.
Fixedlocation circular arc drawing of planar graphs
 J. Graph Alg. & Applications
, 2007
"... In this paper we consider the problem of drawing a planar graph using circular arcs as edges, given a onetoone mapping between the vertices of the graph and a set of points in the plane. If for every edge we have only two possible circular arcs, then a simple reduction to 2SAT yields an O(n2) algo ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
In this paper we consider the problem of drawing a planar graph using circular arcs as edges, given a onetoone mapping between the vertices of the graph and a set of points in the plane. If for every edge we have only two possible circular arcs, then a simple reduction to 2SAT yields an O(n2) algorithm to nd out if a drawing with no crossings can be realized, where n is the number of vertices in the graph. We present an improved O(n7=4polylog n) time algorithm for this problem. For the special case where the possible circular arcs for each edge are of the same length, we present an even more ecient algorithm that runs in O(n3=2polylog n) time. We also consider two related optimization versions of the problem. First we show that minimizing the number of crossings is NPhard. Second we show that maximizing the number of edges that can be realized without crossings is also NPhard. Finally, we show that if we have three or more possible circular arcs per edge, deciding whether a drawing with no crossings can be realized is NPhard.
Open Problems Wiki
"... This project was inspired by the last year's paper on Selected Open Problems in Graph Drawing by Brandenburg et al. (Proc. 11th GD. Vol. 2919 of LNCS. (2003) 515539). While being a very good start, a paper is inherently static and will become outdated. For dynamic content, what open problems ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This project was inspired by the last year's paper on Selected Open Problems in Graph Drawing by Brandenburg et al. (Proc. 11th GD. Vol. 2919 of LNCS. (2003) 515539). While being a very good start, a paper is inherently static and will become outdated. For dynamic content, what open problems (hopefully) are, a website is more appropriate. Keeping such a site uptodate, however, is time consuming and requires good knowledge of recent work. In projects like the free encyclopedia Wikipedia these obstacles are overcome with a collaborative approach: everyone is allowed, and even requested, to contribute his knowledge to the site. The Open Problems Wiki makes use of this paradigm to provide a forum for collecting open problems in graph drawing.
NPCompleteness of Minimal Width Unordered Tree Layout
"... Tree layout has received considerable attention because of its practical importance. Arguably the most common drawing convention is the (ordered) layered tree convention for rooted trees in which the layout is required to preserve the relative order of a node’s children. However, in some application ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Tree layout has received considerable attention because of its practical importance. Arguably the most common drawing convention is the (ordered) layered tree convention for rooted trees in which the layout is required to preserve the relative order of a node’s children. However, in some applications preserving the ordering of children is not important, and considerably more compact layout can be achieved if this requirement is dropped. Here we introduce the unordered layered tree drawing convention for binary rooted trees and show that determining a minimal width drawing for this convention is NPcomplete.
StraightLine Grid Drawings of Planar Graphs with Linear Area
, 2006
"... A straightline grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straightline segment without edge crossings. It is well known that a planar graph of n vertices admits a straightline grid drawing on a g ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
A straightline grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straightline segment without edge crossings. It is well known that a planar graph of n vertices admits a straightline grid drawing on a grid of area O(n 2). A lower bound of Ω(n 2) on the arearequirement for straightline grid drawings of certain planar graphs are also known. In this paper, we introduce a fairly large class of planar graphs which admits a straightline grid drawing on a grid of area O(n). Our new class of planar graphs, which we call “doughnut graphs, ” is a subclass of 5connected planar graphs. We also show several interesting properties of “doughnut graphs ” in this paper.
ORTHOGONAL DRAWINGS OF SERIESPARALLEL GRAPHS WITH MINIMUM BENDS
, 2007
"... In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of G is ca ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of G is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of G. In this paper we give an algorithm to find an optimal orthogonal drawing of any given seriesparallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.