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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 ..."
Abstract

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.
Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
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Cited by 65 (0 self)
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We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
ONLINE PLANARITY TESTING
, 1996
"... The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online plan ..."
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Cited by 26 (2 self)
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The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(log n) time, where n is the current number of vertices of G. The bounds for tests and vertex insertions are worstcase and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.
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.
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 15 (1 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...
Hexagonal Grid Drawings
, 1992
"... In this paper we present a linear algorithm to draw triconnected planar graphs of degree 3 planar on a linearsized hexagonal grid such that in at most one edge are bends. This algorithm can be used to draw this class of graphs planar with straight lines on a n]2 x n]2 grid, improving the best kn ..."
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Cited by 10 (0 self)
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In this paper we present a linear algorithm to draw triconnected planar graphs of degree 3 planar on a linearsized hexagonal grid such that in at most one edge are bends. This algorithm can be used to draw this class of graphs planar with straight lines on a n]2 x n]2 grid, improving the best known grid bounds by a factor 4. We also show how to draw planar graphs of degree at most 3 planar with straight lines such that the minimum angle is >_ r/6, thereby answering a question of Formann et al.