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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 ..."
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Cited by 544 (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.
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 21 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 6 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
On a visibility representation of graphs
 IN [41
, 1996
"... We give a visibility representation of graphs which extends some very wellknown representations considered extensively in the literature. Concretely, the vertices are represented by a collection of parallel hyperrectangles in R n and the visibility is orthogonal to those hyperrectangles. With thi ..."
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Cited by 6 (0 self)
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We give a visibility representation of graphs which extends some very wellknown representations considered extensively in the literature. Concretely, the vertices are represented by a collection of parallel hyperrectangles in R n and the visibility is orthogonal to those hyperrectangles. With this generalization, we can prove that each graph admits a visibility representation. But, it arises the problem of determining the minimum Euclidean space where such representation is possible. We consider this problem for concrete wellknown families of graphs such as planar graphs, complete graphs and complete bipartite graphs.
Further results on bar kvisibility graphs
, 2005
"... A bar visibility representation of a graph G is a collection of horizontal bars in the plane corresponding to the vertices of G such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar kvisibility graph, two vertices ..."
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Cited by 4 (0 self)
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A bar visibility representation of a graph G is a collection of horizontal bars in the plane corresponding to the vertices of G such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar kvisibility graph, two vertices are adjacent if and only if the corresponding bars can be joined by a vertical line segment that intersects at most k other bars. Bar kvisibility graphs were introduced by Dean, Evans, Gethner, Laison, Safari, and Trotter in [3],[4]. In this paper, we present sharp upper bounds on the maximum number of edges in a bar kvisibility graph on n vertices and the largest order of a complete bar kvisibility graph. We also discuss regular bar kvisibility graphs and forbidden induced subgraphs of bar kvisibility graphs.
New Algorithms for Minimizing the Longest Wire Length During Circuit Compaction
, 1995
"... Consider the problem of performing 1dimensional circuit compaction for a layout containing n h horizontal wires and n layout cells. We present new and efficient constraintgraph based algorithms for generating a compacted layout in which either the length of the longest wires or a userspecified t ..."
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Consider the problem of performing 1dimensional circuit compaction for a layout containing n h horizontal wires and n layout cells. We present new and efficient constraintgraph based algorithms for generating a compacted layout in which either the length of the longest wires or a userspecified tradeoff function between the layout width and the longest wire length is minimized. Both algorithms have an O(n h \Delta n log n) running time. The concept employed by our algorithms is that of assigning speeds to the layout cells. Speeds are computed by performing path computations in subgraphs of the constraint graphs. A compacted layout is generated over a number of iterations, with each iteration first determining speeds and then moving the layout elements to the right according to the computed speeds. Each iteration produces a better layout and after at most n \Delta n h iterartions the final layout is produced. Keywords: Analysis of algorithms, circuit layout, compaction, layout width...
Bar kVisibility Graphs
"... Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a onetoone correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if t ..."
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Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a onetoone correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar kvisibility graphs. We obtain an upper bound on the number of edges in a bar kvisibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1visibility graph. We conjecture that bar 1visibility graphs have thickness at most 2.
Unit Rectangle Visibility Graphs
, 2007
"... Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other ..."
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Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other circuit components in computer chip applications. A graph G is a unit rectangle visibility graph (URVG) if its vertices can be represented by closed unit squares in the plane with sides parallel to the axes and pairwise disjoint interiors, in such a way