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33
Simultaneous embedding of a planar graph and its dual on the grid
, 2002
"... Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar ..."
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Cited by 13 (8 self)
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Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straightline segments and the only crossings are between primaldual pairs of edges. We provide a lineartime algorithm that simultaneously embeds a 3connected planar graph and its dual on a (2n −2) ×(2n −2) integer grid, where n is the total number of vertices in the graph and its dual. Furthermore our embedding algorithm satisfies the two natural requirements mentioned above.
Drawing Planar Partitions I: LLDrawings and LHDrawings
, 1998
"... Let a planar graph G = (V; E) and a partition V = A[B of the vertices be given. Can we draw G without edge crossings such that the partition is clearly visible? Such drawings aid to display partitions and cuts as they arise in various applications. In this paper, we first review a number of mode ..."
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Cited by 11 (1 self)
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Let a planar graph G = (V; E) and a partition V = A[B of the vertices be given. Can we draw G without edge crossings such that the partition is clearly visible? Such drawings aid to display partitions and cuts as they arise in various applications. In this paper, we first review a number of models of displaying the partition. Studying two of these models in detail, we provide necessary and sufficient conditions for the existence of a straightline planar drawing, and algorithms to create such drawings, if possible, with area O(n²).
Convex drawings of 3connected plane graphs
 Algorithmica
, 2007
"... We use Schnyder woods of 3connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0 ≤ ∆ ≤ n 2 − 2. The algorithm is a refinement of the fac ..."
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Cited by 10 (0 self)
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We use Schnyder woods of 3connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0 ≤ ∆ ≤ n 2 − 2. The algorithm is a refinement of the facecountingalgorithm, thus, in particular, the size of the grid is at most (f − 2) × (f − 2). The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder’s and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to 7 7
Convex drawings of graphs with nonconvex boundary
 Proc. of WG 2006
, 2006
"... Abstract. In this paper, we study a new problem of finding a convex drawing of graphs with a nonconvex boundary. It is proved that every triconnected plane graph whose boundary is fixed with a starshaped polygon admits a drawing in which every inner facial cycle is drawn as a convex polygon. Such ..."
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Cited by 8 (2 self)
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Abstract. In this paper, we study a new problem of finding a convex drawing of graphs with a nonconvex boundary. It is proved that every triconnected plane graph whose boundary is fixed with a starshaped polygon admits a drawing in which every inner facial cycle is drawn as a convex polygon. Such a drawing, called an innerconvex drawing, can be obtained in linear time. 1
Drawing planar partitions III: Two constrained embedding problems
, 1998
"... In two previous papers, we studied the following problem: Given a planar graph G = (V; E) and a partition V = A [ B of the vertices. Can we draw G without crossing such that the partition is clearly visible? For three models used to display the partition, we developed necessary and sufficient con ..."
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Cited by 8 (1 self)
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In two previous papers, we studied the following problem: Given a planar graph G = (V; E) and a partition V = A [ B of the vertices. Can we draw G without crossing such that the partition is clearly visible? For three models used to display the partition, we developed necessary and sufficient conditions for the existence of such a drawing. The time to test these conditions was dominated by verifying whether there exists a planar embedding of G that satisfies certain additional properties. In this paper, we study how these constrained embedding tests can be done in linear time.
Colored simultaneous geometric embeddings
 In 13th Computing and Combinatorics Conference (COCOON
, 2007
"... Abstract. We introduce the concept of colored simultaneous geometric embeddings as a generalization of simultaneous graph embeddings with and without mapping. We show that there exists a universal pointset of size n for paths colored with two or three colors. We use these results to show that colore ..."
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Cited by 6 (3 self)
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Abstract. We introduce the concept of colored simultaneous geometric embeddings as a generalization of simultaneous graph embeddings with and without mapping. We show that there exists a universal pointset of size n for paths colored with two or three colors. We use these results to show that colored simultaneous geometric embeddings exist for: (1) a 2colored tree together with any number of 2colored paths and (2) a 2colored outerplanar graph together with any number of 2colored paths. We also show that there does not exist a universal pointset of size n for paths colored with five colors. We finally show that the following simultaneous embeddings are not possible: (1) three 6colored cycles, (2) four 6colored paths, and (3) three 9colored paths. 1
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 � �
Convex Drawings of Graphs in Two and Three Dimensions (Preliminary Version)
"... We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) \Theta (3n=2) area under the edge L1 resolution rule; ffl 2D strictly convex grid drawings with O(n 3 ) \Theta O(n 3 ) area under the e ..."
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Cited by 6 (0 self)
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We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) \Theta (3n=2) area under the edge L1 resolution rule; ffl 2D strictly convex grid drawings with O(n 3 ) \Theta O(n 3 ) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) \Theta O(n) area under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers; ffl 3D convex drawings with O(1) \Theta O(1) \Theta O(n) volume under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers. We also show the following lower bounds: ffl For infinitely many nvertex graphs G, if G has a straightline 2D convex drawing in a w \Theta h grid satisfying the edge L1 resolution rule then w;h 5n=6 +\Omega\Gamma20 and w + h 8n=3 + \Omega\Gamma838 ffl For infinitely many boundeddegree triconnected planar graphs G with n ver...
Orthogonal cartograms with few corners per face
, 2010
"... We give an algorithm to create orthogonal drawings of 3connected 3regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 ..."
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Cited by 4 (1 self)
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We give an algorithm to create orthogonal drawings of 3connected 3regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face.
Graph Drawing Algorithm Engineering with AGD
, 2000
"... We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basic ..."
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Cited by 3 (2 self)
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We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basically the same functionality. This enables us to exchange components and experiment with various algorithms and implementations of the same type. We give examples for algorithm engineering with AGD for drawing general nonhierarchical graphs and hierarchical graphs.