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Planar Drawings and Angular Resolution: Algorithms and Bounds (Extended Abstract)
 IN PROC. 2ND ANNU. EUROPEAN SYMPOS. ALGORITHMS
, 1994
"... We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on th ..."
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Cited by 35 (5 self)
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We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on the angular resolution of planar straightline drawings, and show a continuous tradeoff between the area and the angular resolution. We also give lineartime algorithms for constructing planar straightline drawings with high angular resolution for various classes of graphs, such as seriesparallel graphs, outerplanar graphs, and triangulations generated by nested triangles. Our results are obtained by new techniques that make extensive use of geometric constructions.
Checking the Convexity of Polytopes and the Planarity of Subdivisions
, 1998
"... This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivi ..."
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Cited by 24 (7 self)
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This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required.
Characterizing Proximity Trees
, 1996
"... Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, gabriel graph or modified gabriel graph of a set of points in the plane. The characterizations give rise to lineartime algorithms for determining whether a tre ..."
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Cited by 21 (11 self)
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Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, gabriel graph or modified gabriel graph of a set of points in the plane. The characterizations give rise to lineartime algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of gabriel graphs settles the conjectures of Matula and Sokal [19].
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and pro ..."
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Cited by 20 (4 self)
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Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)area planar upward grid drawing that preserves the lefttoright ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...
Lombardi Drawings of Graphs
"... We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each ..."
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Cited by 20 (9 self)
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We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.
Proximity Constraints and Representable Trees
, 1995
"... This paper examines an infinite family of proximity drawings of graphs called open and closed fidrawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the wellknown Gabriel, relative neighborhood and ..."
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Cited by 19 (11 self)
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This paper examines an infinite family of proximity drawings of graphs called open and closed fidrawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the wellknown Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit open fidrawings for 0 fi ! fi ! 1 or closed fidrawings for 0 fi ! fi 1 are given, as well as partial characterizations for other values of fi. For the intervals of fi in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed fidrawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.
Graph Drawing
 Lecture Notes in Computer Science
, 1997
"... INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducte ..."
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Cited by 14 (3 self)
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INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and humancomputer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin
Drawing High Degree Graphs with Low Bend Numbers
 PROC. 4TH SYMPOSIUM ON GRAPH DRAWING (GD'95), LNCS 1027
, 1995
"... We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any respect to the size of the vertices. Naturally, the vertice ..."
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Cited by 14 (2 self)
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We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any respect to the size of the vertices. Naturally, the vertices should be represented by uniformly small squares. In addition we might require that each face should be represented by a nonempty region. This would allow a labeling of the faces. We present an efficient algorithm which provably achieves the bend minimum following these constraints. Omitting the latter requirement we conjecture that the problem becomes NPhard. For that case, we give advices for good approximations. We demonstrate the effectiveness of our approaches giving some interesting examples.
Planar embeddings of graphs with specified edge lengths
, 2007
"... We consider the problem of finding a planar straightline embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see tha ..."
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Cited by 13 (2 self)
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We consider the problem of finding a planar straightline embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NPhard. In contrast, we show that the problem is tractable—indeed, solvable in linear time on a real RAM—for straightline embeddings of planar 3connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NPhard if we consider instead straightline embeddings of planar 3connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.
The Rectangle of Influence Drawability Problem
 Computational Geometry: Theory and Applications
, 1997
"... Motivated by rectangular visibility and graph drawing applications, we study the problem of characterizing classes of graphs that admit rectangle of influence drawings. We consider several classes of graphs and show, for each class, that testing whether a graph G has a rectangle of influence draw ..."
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Cited by 12 (3 self)
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Motivated by rectangular visibility and graph drawing applications, we study the problem of characterizing classes of graphs that admit rectangle of influence drawings. We consider several classes of graphs and show, for each class, that testing whether a graph G has a rectangle of influence drawing can be done in O(n) time, where n is the number of vertices of G. If the test for G is affirmative, we show how to construct a rectangle of influence drawing of G. All the drawing algorithms can be implemented so that they (1) produce drawings with all vertices placed at intersection points of an integer grid of size O(n 2 ), (2) perform arithmetic operations on integers only, and (3) run in O(n) time, where n is the number of vertices of the input graph. 1 Introduction A proximity drawing of a graph is a straightline drawing (vertices are represented by points and edges by straightline segments) where the points representing adjacent vertices are deemed to be close according t...