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98
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 34 (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.
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
FULLY DYNAMIC POINT LOCATION IN A MONOTONE SUBDIVISION
, 1989
"... In this paper a dynamic technique for locating a point in a monotone planar subdivision, whose current number of vertices is n, is presented. The (complete set of) update operations are insertion of a point on an edge and of a chain of edges between two vertices, and their reverse operations. The d ..."
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Cited by 23 (7 self)
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In this paper a dynamic technique for locating a point in a monotone planar subdivision, whose current number of vertices is n, is presented. The (complete set of) update operations are insertion of a point on an edge and of a chain of edges between two vertices, and their reverse operations. The data structure uses space O(n). The query time is O(log n), the time for insertion/deletion of a point is O(log n), and the time for insertion/deletion of a chain with k edges is O(log n + k), all worstcase. The technique is conceptually a special case of the chain method of Lee and Preparata and uses the same query algorithm. The emergence of full dynamic capabilities is afforded by a subtle choice of the chain set (separators), which induces a total order on the set of regions of the planar subdivision.
Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 23 (11 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices.
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
A LeftFirst Search Algorithm for Planar Graphs
 Discrete Computational Geometry
, 1995
"... We give an O(jV (G)j) time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we o ..."
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Cited by 20 (4 self)
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We give an O(jV (G)j) time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovic. The edges of any maximal bipartite plane graph G with outer face bwb w can be colored by two colors such that the color classes form spanning trees of G b and G b , respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear time algorithm for constructing bipolar orientations of 2connected plane graphs.
Radial Level Planarity Testing and Embedding in Linear Time
 Journal of Graph Algorithms and Applications
, 2005
"... A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines an ..."
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Cited by 19 (9 self)
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A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are level nonplanar biconnected components. Our main results are linear time algorithms for radial level planarity testing and for computing an embedding. We introduce PQRtrees as a new data structure where Rnodes and associated templates for their manipulation are introduced to deal with rings. Our algorithms extend level planarity testing and embedding algorithms which use PQtrees.
An O(m log n)Time Algorithm for the Maximal Planar Subgraph Problem
, 1993
"... Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C1 ..."
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Cited by 17 (0 self)
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Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [15], Wu defined the problem of planar graphs in terms of the following four subproblems: ################## 1 This work was partly supported by ThomsonCSF/DSE and by the National Science Foundation under grant CCR9002428. 2. Research at Princeton University partially supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, grant NSFSTC8809648, and the Office of Naval Research, contract N0001487K0467.    2  P1. Decide whether a connected graph G is planar. P2. Find a minimal set of edges the removal of which will render the remaining part of G planar. P3. Gi...
Representation of Planar Graphs By Segments
, 1994
"... Given any bipartite planar graph G, one can assign vertical and horizontal segments to its vertices so that (a) no two of them have an interior point in common, (b) two segments have a point in common if and only if the corresponding vertices are adjacent in G. ..."
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Cited by 16 (1 self)
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Given any bipartite planar graph G, one can assign vertical and horizontal segments to its vertices so that (a) no two of them have an interior point in common, (b) two segments have a point in common if and only if the corresponding vertices are adjacent in G.
Randomized Graph Drawing with HeavyDuty Preprocessing
 In: AVI ’94: Proceedings of the Workshop on Advanced Visual Interfaces
, 1994
"... : We present a graph drawing system for general undirected graphs with straightline edges. It carries out a rather complex set of preprocessing steps, designed to produce a topologically good, but not necessarily nicelooking layout, which is then subjected to Davidson and Harel's simulated anneali ..."
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Cited by 14 (1 self)
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: We present a graph drawing system for general undirected graphs with straightline edges. It carries out a rather complex set of preprocessing steps, designed to produce a topologically good, but not necessarily nicelooking layout, which is then subjected to Davidson and Harel's simulated annealing beautification algorithm. The intermediate layout is planar for planar graphs and attempts to come close to planar for nonplanar graphs. The system's results are significantly better, and much faster, than what the annealing approach is able to achieve on its own. 1 Introduction A large amount of work on the problem of graph layout has been carried out in recent years, resulting in a number of sophisticated and powerful algorithms. An extensive and detailed survey can be found in [BETT93]. Many of the approaches taken are limited to special cases of graphs, such as trees or planar graphs; others concentrate on special kinds of layouts, such as rectilinear grid drawings, or convex drawin...