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A Simple Linear Time Algorithm for Proper Box Rectangular Drawing of Plane Graphs
 Journal of Algorithms
, 2000
"... In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is dra ..."
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In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...
Balanced VertexOrderings of Graphs
, 2002
"... We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains N ..."
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We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains NPhard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertexordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we
Orthogonal drawings with few layers
 PROC. 9TH INTERNATIONAL SYMP. ON GRAPH DRAWING (GD '01
, 2002
"... In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very smal ..."
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In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain pointdrawings with 3layers and 4 bends per edge, and algorithms to obtain boxdrawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds.
Orthogonal Drawings Based On The Stratification Of Planar Graphs
, 2000
"... Several algorithms have been proposed to draw planar graphs using 2visibility and Kandinsky Models. Here, we propose three new algorithms implementing these models in linear time using small grid sizes and few bends. These algorithms are all based on the construction of a particular layered spannin ..."
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Several algorithms have been proposed to draw planar graphs using 2visibility and Kandinsky Models. Here, we propose three new algorithms implementing these models in linear time using small grid sizes and few bends. These algorithms are all based on the construction of a particular layered spanning tree called Stratification. A linear time algorithm that computes a stratification is also presented.
New Lower Bounds For Orthogonal Drawings
 J. GRAPH ALGORITHMS APPL
, 1998
"... An orthogonal drawing of a graph is an embedding of the graph in the twodimensional grid such that edges are routed along gridlines. In this paper we explore lower bounds for orthogonal graph drawings. We prove lower bounds on the number of bends and, when crossings are not allowed, also lower bou ..."
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An orthogonal drawing of a graph is an embedding of the graph in the twodimensional grid such that edges are routed along gridlines. In this paper we explore lower bounds for orthogonal graph drawings. We prove lower bounds on the number of bends and, when crossings are not allowed, also lower bounds on the size of the grid.
Smooth Orthogonal Layouts
"... Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axisaligned line segments, in smooth orthogonal layouts every edge is made of axisaligned segments and circular arcs with common tange ..."
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Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axisaligned line segments, in smooth orthogonal layouts every edge is made of axisaligned segments and circular arcs with common tangents. Our goal is to create such layouts with low edge complexity, measured by the number of line and circular arc segments. We show that every biconnected 4planar graph has a smooth orthogonal layout with edge complexity 3. If the input graph has a complexity2 traditional orthogonal layout we can transform it into a smooth complexity2 layout. Using the Kandinsky model for removing the degree restriction, we show that any planar graph has a smooth complexity2 layout. 1
Some Applications of Orderly Spanning Trees in Graph Drawing
 IN PROCEEDINGS OF THE 10TH INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING, LNCS 2528
, 2002
"... Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spa ..."
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Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spanning trees of plane graphs are investigated. Our first
Visualizing Software for Telecommunication Services
"... An active research area in telecommunications concerns how to specify and control the addition of new services, such as call waiting or instant messaging, into existing software. One approach is to rely on a componentbased architecture such as Distributed Feature Composition (DFC), by which a new s ..."
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An active research area in telecommunications concerns how to specify and control the addition of new services, such as call waiting or instant messaging, into existing software. One approach is to rely on a componentbased architecture such as Distributed Feature Composition (DFC), by which a new service can be specified as a composition of primitive features over time. Formally, a communication episode is represented by a dynamic graph of software feature boxes, called a usage. This serves as the fundamental model for how services are invoked and how they interact with other services. This paper, after providing some background on DFC, discusses a technique for visualizing the usages which arise through DFC specifications. With the visualization, users can monitor and validate service protocols and feature interactions in real time or through playback logs. The principal display component uses a novel variation of forcedirected layouts for undirected graphs. The resulting graphical interface has become a principal tool for developers building services using DFC.
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...
Minimising the Number of Bends and Volume in 3Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout
, 2004
"... A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3d ..."
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A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, socalled diagonal drawings. This vertexlayout strategy was introduced in the 3BENDS algorithm of Eades et al. [Discrete Applied Math. 103:5587, 2000]. We show that minimising the number of bends in a diagonal drawing of a given graph is NPhard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal in linear time. Using two heuristics for determining this vertexordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the abovementioned 3BENDS algorithm, produces 3bend drawings with o(n ) volume, which is the best known upper bound for the volume of 3dimensional orthogonal graph drawings with at most three bends per edge.