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Orthogonal Drawings of Plane Graphs without Bends
, 2003
"... In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. Every plane graph of the maximum degree at most four has an orthogonal dra ..."
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In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. Every plane graph of the maximum degree at most four has an orthogonal drawing, but may need bends. A simple necessary and sufficient condition has not been known for a plane graph to have an orthogonal drawing without bends. In this paper we obtain a necessary and sufficient condition for a plane graph G of the maximum degree three to have an orthogonal drawing without bends. We also give a lineartime algorithm to find such a drawing of G if it exists.
Cyberspace geography visualization  Mapping the WorldWide Web to help people nd their way in cyberspace, heiwww.unige.ch/girardin/cgv
 of the WorldWide Web, heiwww.unige.ch/girardin/ cgv/www5/index.html HIPPNER
, 1995
"... Abstract As cyberspace becomes an integral part of our daily life, its mastering becomes harder. To help, cyberspace can be represented by resources arranged in a multidimensional space. With geographical maps to exhibit the topology of this virtual space, people can have a better visual understandi ..."
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Abstract As cyberspace becomes an integral part of our daily life, its mastering becomes harder. To help, cyberspace can be represented by resources arranged in a multidimensional space. With geographical maps to exhibit the topology of this virtual space, people can have a better visual understanding. In this paper, methods focusing on the construction of lower dimension representations of this space are examined and illustrated with the WorldWide Web. It is expected that this work will contribute to addressing issues of navigation in cyberspace and, especially, avoiding the lostincyberspace syndrome. Résumé Alors que le cyberspace envahit notre vie quotidienne, sa maîtrise devient de plus en plus complexe. On peut l’imaginer comme un ensemble de ressources arrangées dans un espace multidimensionnel. En utilisant des cartes géographiques pour représente la topologie virtuelle de cet espace, on arrive à mieux le comprendre, le cerner. Dans ce papier, des méthodes se concentrant sur la construction de représentations à dimensions réduites sont étudiées en les appliquant au WorldWide Web. On espère que ce travail contribuera à résoudre les problèmes de navigation dans ce monde virtuel et en particulier à éviter de s’y perdre. Ubersicht In einer Zeit, in der der Cyberspace ein integraler Bestandteil unseres täglichen Lebens wird, wird seine Beherrschung zunehmend schwieriger. Zur Erleichterung kann Cyberspace anhand von Quellen, angeordnet in einem multidimensionalen Raum, dargestellt werden. Mit geographischen Karten, die die Topologie dieses künstlichen Raumes aufzeigen, kann das visuelle Verständnis verbessert werden. In dieser Arbeit werden Methoden zur Konstruktion von Darstellungen mit niedriger Dimension dieses Raumes untersucht und anhand des WorldWide Web verdeutlicht.
Efficient Algorithms for Drawing Planar Graphs
, 1999
"... x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ................. ..."
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x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ............................ 6 1.2.4 Orthogonal drawings . . ........................... 7 1.2.5 Grid drawings ................................ 8 1.3 Properties of Drawings ................................ 9 1.4 Scope of this Thesis .................................. 10 1.4.1 Rectangular drawings . . . ......................... 11 1.4.2 Orthogonal drawings . . ........................... 12 1.4.3 Boxrectangular drawings ........................... 14 1.4.4 Convex drawings . . ............................. 16 1.5 Summary ....................................... 16 2 Preliminaries 20 2.1 Basic Terminology .................................. 20 2.1.1 Graphs and Multigraphs ........................... 20 i CO...
On the Volume and Resolution of 3Dimensional Convex Graph Drawing (Extended Abstract)
"... We address the problem of drawing a 3connected planar graph as a convex polyhedron in R³. We give an efficient algorithm for producing such a realization using O(n) volume under the vertexresolution rule. Each vertex in the drawing resulting from this method is guaranteed to need no more than ..."
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We address the problem of drawing a 3connected planar graph as a convex polyhedron in R³. We give an efficient algorithm for producing such a realization using O(n) volume under the vertexresolution rule. Each vertex in the drawing resulting from this method is guaranteed to need no more than O(n log n) bits to represent (as a pair of rational numbers). This solves an open problem of Cohen, Eades, Lin, and Ruskey. We also show that under the angularresolution rule drawing a 3connected planar graph as a convex polyhedron in R³ requires at least exponential volume in the worst case.
ORTHOGONAL DRAWINGS OF SERIESPARALLEL GRAPHS WITH MINIMUM BENDS ∗
"... Abstract. 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 ..."
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Abstract. 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.
On a Class of Planar Graphs with . . .
, 2009
"... 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 ..."
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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). We give a lineartime algorithm to find such a drawing. Our new class of planar graphs, which we call “doughnut graphs, ” is a subclass of 5connected planar graphs. We show several interesting properties of “doughnut graphs” in this paper. One can easily observe that any spanning subgraph of a “doughnut graph” also admits a straightline grid drawing with linear area. But the recognition of a spanning subgraph of a “doughnut graph” seems to be a nontrivial problem, since the recognition of a spanning subgraph of a given graph is an NPcomplete problem in general. We establish a necessary and sufficient condition for a 4connected planar graph G to be a spanning subgraph of a “doughnut graph.” We also give a lineartime algorithm to augment a 4connected planar graph G to a “doughnut graph” if G satisfies the necessary and sufficient condition.
Every fourcolorable graph is isomorphic to a subgraph of the Visibility Graph of the Integer Lattice
"... We prove that a graph is 4colorable if and only if it can be drawn with vertices in the integer lattice, using as edges only line segments not containing a third point of the lattice. 1 ..."
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We prove that a graph is 4colorable if and only if it can be drawn with vertices in the integer lattice, using as edges only line segments not containing a third point of the lattice. 1
MinimumArea Drawings of Plane 3Trees (Extended Abstract)
"... A straightline grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straightline segment. The area of such a drawing is the area of the smallest axisaligned rectangle on the grid which encloses the draw ..."
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A straightline grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straightline segment. The area of such a drawing is the area of the smallest axisaligned rectangle on the grid which encloses the drawing. A minimumarea drawing of a plane graph G is a straightline grid drawing of G where the area of the drawing is the minimum. Although it is NPhard to find minimumarea drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3trees in polynomial time. Furthermore, we show a ⌊2n n 3 − 1 ⌋ × 2⌈ 3 ⌉ lower bound for the area of a straightline grid drawing of a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1)
More Canonical Ordering
, 2010
"... Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new appr ..."
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Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. Further, we discuss duality aspects and relations to Schnyder woods. Submitted: