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27
StraightLine Drawings on Restricted Integer Grids in Two and Three Dimensions (Extended Abstract)
, 2002
"... This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straightline crossingfree drawings with vertices located at the grid points of phi? We characterize the trees t ..."
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Cited by 38 (4 self)
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This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straightline crossingfree drawings with vertices located at the grid points of phi? We characterize the trees that can be drawn on a two dimensional c * n × k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossingfree with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal  it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossingfree straight line 3d drawings in linear volume for a nontrivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.
Convex Drawings of Graphs in Two and Three Dimensions
, 1996
"... 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) × (3n/2) area under the edge L 1 resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution ru ..."
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Cited by 29 (10 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) × (3n/2) area under the edge L 1 resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) × 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) × O(1) × O(n) volume under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers. We also
Planar Drawings of Plane Graphs
, 2000
"... this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results. ..."
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Cited by 13 (3 self)
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this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.
Octagonal drawings of plane graphs with prescribed face areas
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 2009
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Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segme ..."
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Cited by 10 (3 self)
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We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
A Simple Linear Time Algorithm for Proper Box Rectangular Drawing of Plane Graphs
 Journal of Algorithms
, 2000
"... In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is dra ..."
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Cited by 7 (0 self)
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In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...
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.