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12
A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows ..."
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Cited by 520 (40 self)
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Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
Level Planar Embedding in Linear Time
, 1999
"... A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be ..."
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Cited by 20 (0 self)
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A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i, all v V i are drawn on the line li  {(x, k  i) ] x ), the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to
Constraints in graph drawing algorithms
 Constraints
, 1998
"... Abstract. Graphs are widely used for information visualization purposes, since they provide a natural and intuitive representation of complex abstract structures. The automatic generation of drawings of graphs has applications a variety of fields such as software engineering, database systems, and g ..."
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Cited by 15 (0 self)
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Abstract. Graphs are widely used for information visualization purposes, since they provide a natural and intuitive representation of complex abstract structures. The automatic generation of drawings of graphs has applications a variety of fields such as software engineering, database systems, and graphical user interfaces. In this paper, we survey algorithmic techniques for graph drawing that support the expression and satisfaction of userdefined constraints. 1.
Visualizing graphs  a generalized view
 In Proceedings of the conference on Information Visualization (IV’06
, 2006
"... The visualization of graphs has proven to be very useful for exploring structures in different application domains. However, in certain fields of computer science, graph visualization is understood and focused quite differently. While ”graph drawing ” focuses on optimized layouts for nodelinkrepres ..."
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Cited by 12 (2 self)
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The visualization of graphs has proven to be very useful for exploring structures in different application domains. However, in certain fields of computer science, graph visualization is understood and focused quite differently. While ”graph drawing ” focuses on optimized layouts for nodelinkrepresentations of networks, ”information visualization” prefers to work on hierarchies focusing on very large structures, different views and interactivity. This paper gives a systematic view of the problem of graph visualization by combining both approaches. We will introduce a general view of different visualization methods as well as describe occurring problems and discuss basic constraints. These will be used to propose a visualization framework for graphs, whose development motivated this paper.
Evaluating monotone circuits on cylinders, planes, and torii
 In Proc. 23rd Symposium on Theoretical Aspects of Computing (STACS), Lecture Notes in Computer Science
, 2006
"... Abstract. We revisit monotone planar circuits MPCVP, with special attention to circuits with cylindrical embeddings. MPCVP is known to be in NC 3 in general, and in LogDCFL for the special case of upward stratified circuits. We characterize cylindricality, which is stronger than planarity but strict ..."
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Cited by 10 (2 self)
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Abstract. We revisit monotone planar circuits MPCVP, with special attention to circuits with cylindrical embeddings. MPCVP is known to be in NC 3 in general, and in LogDCFL for the special case of upward stratified circuits. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that monotone circuits with embeddings that are stratified cylindrical, cylindrical, planar oneinputface and focused can be evaluated in LogDCFL, AC 1 (LogDCFL), LogCFL and AC 1 (LogDCFL) respectively. We note that the NC 3 algorithm for general MPCVP is in AC 1 (LogCFL) =SAC 2.Finally, we show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. 1
Tessellation and Visibility Representations of Maps on the Torus
"... The model of the torus as a parallelogram in the plane with opposite sides identified enables us to define two families of parallel lines and to tessellate the torus, then to associate to each tessellation a toroidal map with an upward drawing. It is proved that a toroidal map has a tessellation ..."
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Cited by 5 (0 self)
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The model of the torus as a parallelogram in the plane with opposite sides identified enables us to define two families of parallel lines and to tessellate the torus, then to associate to each tessellation a toroidal map with an upward drawing. It is proved that a toroidal map has a tessellation representation if and only if its universal cover is 2connected. Those graphs that admit such an embedding in the torus are characterized. 1 Introduction Given a graph G, let V (G) be the set of vertices of G, E(G) the set of edges of G. For A ` V (G) we denote by E(A) the set of edges of G with both ends in A. A map M on a surface \Sigma is a connected graph G together with a 2cell embedding of G in \Sigma. Two maps are equivalent if there is a homeomorphism 1 Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1021010194. 2 This work was partially supported by the ESPRIT Basic Research Action No. 7141 (ALCOM II). 1 of \Sigma mapping...
Regular Labelings and Geometric Structures
, 2010
"... Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the ..."
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Cited by 2 (1 self)
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Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.
IMPROVED UPPER BOUNDS IN NC FOR MONOTONE PLANAR CIRCUIT VALUE AND SOME RESTRICTIONS AND GENERALIZATIONS
"... and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we reexamine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward ..."
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and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we reexamine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC 1 (LogDCFL), while monotone circuits with oneinputface planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC 1 (LogDCFL). We reexamine the NC 3 algorithm for general MPCVP, and note that it is in AC 1 (LogCFL) = SAC 2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar nonmonotone circuits with polylogarithmic negationheight can be evaluated in NC.
Testing Bipartiteness of Geometric Intersection Graphs
, 2003
"... Abstract. We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of balls in R d, in timeO(n log n). More generally we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geo ..."
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Abstract. We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of balls in R d, in timeO(n log n). More generally we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric objects. For unit balls in R d, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and hence is unlikely to be solved as efficiently as bipartiteness testing. For line segments or planar disks, testing kcolorability of intersection graphs for k> 2 is NPcomplete. 1