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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 81 (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 78 (14 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 33 (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.
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 6 (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
Layered Drawings of Digraphs
, 2000
"... Contents 1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.5 1.5.1 1.5.2 1.6 1.7 1.7.1 1.7.2 1.7.3 1.7.4 1.7.5 Layered Drawings of Digraphs 01iver Basteft and Christian Matuszewski Introduction ........................... Cycle Re ..."
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Cited by 5 (0 self)
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Contents 1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.5 1.5.1 1.5.2 1.6 1.7 1.7.1 1.7.2 1.7.3 1.7.4 1.7.5 Layered Drawings of Digraphs 01iver Basteft and Christian Matuszewski Introduction ........................... Cycle Removal .......................... Fast Heuristics .......................... An Enhanced Greedy Heuristic ................. A Randomized Algorithm .................... An Exact Algorithm ....................... Layer Assignment ........................ Layerings for General Graphs .................. Minimizing the Height ..................... Layerings with Given Width .................. Minimizing the Total Edge Span ................ Crossing Reduction ........................ The LayerbyLayer Sweep .................... One Sided Crossing Minimization ................ Klayer Crossing Minimization ................. Dense Graphs and Edge Concentration ............. Horizontal
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...
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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Cited by 1 (0 self)
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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.
Upward Planarity Testing via SAT
"... Abstract. A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing ycoordinates. The problem to decide whether a graph is upward planar or not is NPcomplete in general, and while special graph classes a ..."
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Cited by 1 (1 self)
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Abstract. A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing ycoordinates. The problem to decide whether a graph is upward planar or not is NPcomplete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branchandbound algorithm over the graph’s triconnectivity structure which was able to solve sparse graphs. In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively. 1
On the Complexity of Finding . . .
, 2007
"... Upward planarity is a widely investigated topic in graph theory and graph drawing. A planar digraph is said to be upward planar if it admits a planar drawing where all edges are drawn as curves monotonically increasing in the upward direction. It is known that testing whether a planar digraph is upw ..."
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Upward planarity is a widely investigated topic in graph theory and graph drawing. A planar digraph is said to be upward planar if it admits a planar drawing where all edges are drawn as curves monotonically increasing in the upward direction. It is known that testing whether a planar digraph is upward planar is NPComplete in general. However, the upward planarity testing problem can be solved in polynomial time if the planar embedding of the digraph is fixed. This motivates the following question: Given an embedded planar digraph G, how is difficult to find an embedding preserving subgraph of G that is upward planar and whose number of edges is maximum? We prove that this problem is NPHard. This negative result motivates the study of polynomialtime algorithms for the computation of maximal (not necessarily maximum) upward planar subgraphs as well as the design of exact algorithms that perform well in practice. As a consequence of our proof technique, we also show that the problem of extracting a maximum bimodal subgraph from an embedded planar digraph is NPHard.
Upward Planar Drawings . . .
, 2006
"... In this paper we present a new characterization of switchregular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popula ..."
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In this paper we present a new characterization of switchregular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popular approach described by Bertolazzi, Di Battista, Liotta, and Mannino, our algorithm computes drawings that are significantly better in terms of total edge length and aspect ratio, especially for lowdensity digraphs. Also, we experimentally prove that the running time of the drawing process is reduced in most cases.