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Convex Grid Drawings of 3Connected Planar Graphs
, 1994
"... We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane gr ..."
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Cited by 37 (7 self)
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We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane graph G (with n 3), finds such a straightline convex embedding of G into a (n \Gamma 2) \Theta (n \Gamma 2) grid. 1 Introduction In this paper we consider the problem of aesthetic drawing of plane graphs, that is, planar graphs that are already embedded in the plane. What is exactly an aesthetic drawing is not precisely defined and, depending on the application, different criteria have been used. In this paper we concentrate on the two following criteria: (a) edges should be represented by straightline segments, and (b) faces should be drawn as convex polygons. F'ary [6], Stein [14] and Wagner [18] showed, independently, that each planar graph can be drawn in the plane in such a way that ...
Tutte’s barycenter method applied to isotopies
 Computational Geometry: Theory and Applications
, 2001
"... This paper is concerned with applications of Tutte’s barycentric embedding theorem (Proc. London Math. Soc. 13 (1963), 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremo ..."
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Cited by 9 (0 self)
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This paper is concerned with applications of Tutte’s barycentric embedding theorem (Proc. London Math. Soc. 13 (1963), 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremona’s theorem; it also provides a counterexample showing that the analogue of Tutte’s theorem in dimension 3 is false.
OnLine Convex Planarity Testing
, 1995
"... An important class of planar straightline drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph i ..."
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Cited by 6 (3 self)
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An important class of planar straightline drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph is subject to online insertions of vertices and edges. We present online algorithms for convex planarity testing with the following performance, where t denotes the number of vertices of the graph: convex planarity testing and insertion of vertices take 0(1) worstcase tinhe, insertion of edges takes 0(log n) amortized tinhe, and the space requirement of the data structure is O(n). Furthermore, we give a new combinatorial characterization of convex planar graphs.
Incremental Convex Planarity Testing
, 2001
"... An important class of planar straightline drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decompos ..."
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An important class of planar straightline drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of testing convex planarity in an incremental environment, where a biconnected planar graph is subject to online insertions of vertices and edges. We present a data structure for the online incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity testing takes O(1) worstcase time, insertion of vertices takes O(log n) worstcase time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n).
On Drawing a Graph Convexly in the Plane (Extended Abstract)
"... ) ? Hristo N. Djidjev Department of Computer Science, Rice University, Hoston, TX 77251, USA Abstract. Let G be a planar graph and H be a subgraph of G. Given any convex drawing of H, we investigate the problem of how to extend the drawing of H to a convex drawing of G. We obtain a necessary and s ..."
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) ? Hristo N. Djidjev Department of Computer Science, Rice University, Hoston, TX 77251, USA Abstract. Let G be a planar graph and H be a subgraph of G. Given any convex drawing of H, we investigate the problem of how to extend the drawing of H to a convex drawing of G. We obtain a necessary and sufficient condition for the existence and a linear algorithm for the construction of such an extension. Our results and their corollaries generalize previous theoretical and algorithmic results of Tutte, Thomassen, Chiba, Yamanouchi, and Nishizeki. 1 Introduction The problem of embedding of a graph in the plane so that the resulting drawing has nice geometric properties has received recently significant attention. This is due to the large number of applications including circuit and VLSI design, algorithm animation, information systems design and analysis. The reader is referred to [1] for annotated bibliography on graph drawings. The first lineartime algorithm for testing a graph for plan...