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12
On the Computational Complexity of Upward and Rectilinear Planarity Testing (Extended Abstract)
, 1994
"... A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical se ..."
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Cited by 98 (4 self)
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A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment, and no two edges cross. Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures. In this paper we show that upward planarity testing and rectilinear planarity testing are NPcomplete problems. We also show that it is NPhard to approximate the minimum number of bends in a planar orthogonal drawing of an nvertex graph with an O(n 1\Gammaffl ) error, for any ffl ? 0.
Upward Planarity Testing
 SIAM Journal on Computing
, 1995
"... Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upw ..."
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Cited by 82 (13 self)
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Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and singlesource digraphs. We also sketch the proof of NPcompleteness of upward planarity testing.
Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in softwar ..."
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Cited by 38 (4 self)
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Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
Upward Planar Drawing of Single Source Acyclic Digraphs
, 1990
"... A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We ..."
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Cited by 17 (0 self)
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A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given singlesource acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...
A parameterized algorithm for upward planarity testing
 In Annual European Symposium on Algorithms (Proc. ESA ’04
, 2004
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which eac ..."
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Cited by 7 (0 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NPcomplete. Parameterized complexity is a technique used to find efficient algorithms for hard
Classification of Planar Upward Embedding
"... We consider planar upward drawings of directed graphs on arbitrary surfaces where the upward direction is defined by a vector field. This generalizes earlier approaches using surfaces with a fixed embedding in R 3 and introduces new classes of planar upward drawable graphs, where some of them even ..."
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Cited by 3 (3 self)
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We consider planar upward drawings of directed graphs on arbitrary surfaces where the upward direction is defined by a vector field. This generalizes earlier approaches using surfaces with a fixed embedding in R 3 and introduces new classes of planar upward drawable graphs, where some of them even allow cycles. Our approach leads to a classification of planar upward embeddability. In particular, we show the coincidence of the classes of planar upward drawable graphs on the sphere and on the standing cylinder. These classes coincide with the classes of planar upward drawable graphs with a homogeneous field on a cylinder and with a radial field in the plane. A cyclic field in the plane introduces the new class RUP of upward drawable graphs, which can be embedded on a rolling cylinder. We establish strict inclusions for planar upward drawability on the plane, the sphere, the rolling cylinder, and the torus, even for acyclic graphs. Finally, upward drawability remains NPhard for the standing cylinder and the torus; for the cylinder this was left as an open problem by Limaye et al.
Where to Draw the Line
, 1996
"... Graph Drawing (also known as Graph Visualization) tackles the problem of representing graphs on a visual medium such as computer screen, printer etc. Many applications such as software engineering, data base design, project planning, VLSI design, multimedia etc., have data structures that can be rep ..."
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Cited by 2 (0 self)
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Graph Drawing (also known as Graph Visualization) tackles the problem of representing graphs on a visual medium such as computer screen, printer etc. Many applications such as software engineering, data base design, project planning, VLSI design, multimedia etc., have data structures that can be represented as graphs. With the ever increasing complexity of these and new applications, and availability of hardware supporting visualization, the area of graph drawing is increasingly getting more attention from both practitioners and researchers. In a typical drawing of a graph, the vertices are represented as symbols such as circles, dots or boxes, etc., and the edges are drawn as continuous curves joining their end points. Often, the edges are simply drawn as (straight or poly) lines joining their end points (and hence the title of this thesis), followed by an optional transformation into smooth curves. The goal of research in graph drawing is to develop techniques for constructing good...
Upward threedimensional grid drawings of graphs. arXiv.org math.CO/0510051
, 2005
"... Abstract. A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce threedimensional grid drawings with small bounding box volume. Our first ..."
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Cited by 2 (2 self)
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Abstract. A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce threedimensional grid drawings with small bounding box volume. Our first main result is that every nvertex graph with bounded degeneracy has a threedimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A threedimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the zdirection. We prove that every dag has an upward threedimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every ccolourable dag (c constant) has an upward threedimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward threedimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward threedimensional grid drawings with bends in the edges. 1.
The Duals of Upward Planar Graphs on Cylinders
"... We consider directed planar graphs with an upward planar drawing on the rolling and standing cylinders. These classes extend the upward planar graphs in the plane. Here, we address the dual graphs. Our main result is a combinatorial characterization of these sets of upward planar graphs. It basical ..."
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We consider directed planar graphs with an upward planar drawing on the rolling and standing cylinders. These classes extend the upward planar graphs in the plane. Here, we address the dual graphs. Our main result is a combinatorial characterization of these sets of upward planar graphs. It basically shows that the roles of the standing and the rolling cylinders are interchanged for their duals.
Building blocks of upward planar digraphs
 Proc. GD’04, volume 3383 of LNCS
, 2005
"... The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if ..."
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The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if its blocks admit upward planar drawings with certain properties. We also show how to test if a block of a digraph admits an upward planar drawing with the aforementioned properties.