M. Chrobak, and S. Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29--54, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(involving a function of n) is given and we consider what the graphs are whose drawings t that restricted grid. For example, it is well known that there are families of graphs that require ) area to be drawn in the plane, the canonical example being a sequence of n=3 nested triangles (see [12, 8, 38]) Such graphs can be drawn on the surface of a three dimensional triangular prism of linear volume and using integer coordinates. Thus a natural question is whether there exist speci c restrictions of the 3D integer grid of linear volume that can support straight line crossing free drawings of ....

M. Chrobak, and S. Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29-54, 1998.

....of specified dimensions is given and we consider which graphs have drawings that fit that restricted grid. For example, it is well known that there are families of graphs that require Omega ) area to be drawn in the plane, the canonical example being a sequence of n=3 nested triangles (see [8, 5, 19]) Such graphs can be drawn on the surface of a 3 dimensional triangular prism of linear volume and using integer coordinates. Thus a natural question is whether there exist specific restrictions of the 3d integer grid of linear volume that can support straight line crossing free drawings of ....

Marek Chrobak and Shin ichi Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29--54, 1998.

....were first defined by de Fraysseix, Pach and Pollack [11] for maximal planar graphs and later extended by Kant [29] to triconnected planar graphs. They have been successfully applied to the construction of various types of planar drawings (straight line, orthogonal, and polyline) see, e.g. [8, 11, 29]) Schnyder [37] defines realizers of maximal planar graphs in his study of the order dintension of planar graphs, and shows their application to planar straight line drawings [38] The construction of realizers of maximal planar graphs can also be efficiently parallelized [20] Brightwell and ....

M. Chrobak and S. Nakano. Minimum-width grid drawings of plane graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '9), volume 894 of Lecture Notes in Computer' Science, pages 104 110. Springer-Verlag, 1995.

....grid of speci ed dimensions is given and we consider what the graphs are whose drawings t that restricted grid. For example, it is wellknown that there are families of graphs that require (n 2 ) area to be drawn in the plane, the canonical example being a sequence of n=3 nested triangles (see [8, 5, 19]) Such graphs can be drawn on the surface of a 3 dimensional triangular prism of linear volume and using integer coordinates. Thus a natural question is whether there exist speci c restrictions of the 3d integer grid of linear volume that can support straightline crossing free drawings of ....

Marek Chrobak and Shin ichi Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29-54, 1998.

....grid of speci ed dimensions is given and we consider what the graphs are whose drawings t that restricted grid. For example, it is well known that there are families of graphs that require (n 2 ) area to be drawn in the plane, the canonical example being a sequence of n=3 nested triangles (see [8, 5, 19]) Such graphs can be drawn on the surface of a 3 dimensional triangular prism of linear volume and using integer coordinates. Thus a natural question is whether there exist speci c restrictions of the 3d integer grid of linear volume that can support straight line crossing free drawings of ....

Marek Chrobak and Shin ichi Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29-54, 1998.

....rectangle. The grid size is usually described as W H . The grid size is sometimes described by the half perimeter W H or the area W H of the grid. It is a very challenging problem to draw a plane graph on a grid of the minimum size. In recent years, several works are devoted to this field [FPP90, Sch90, CN98]; for example, any plane graph with n vertices has a straight line drawing on a grid of area W H # (n 1) 2 . 1.3 Properties of Drawings There are infinitely many drawings for a graph. In drawing a graph, we would like to take into account a variety of properties. For example, we may be ....

M. Chrobak and S. Nakano, Minimum-width grid drawings of plane graphs, Comp. Geom. Theory Appl., 11 (1998), pp. 29-54. pp. 104-110.

....were first defined by de Fraysseix, Pach and Pollack [11] for maximal planar graphs and later extended by Kant [29] to triconnected planar graphs. They have been successfully applied to the construction of various types of planar drawings (straight line, orthogonal, and polyline) see, e.g. [8, 11, 29]) Schnyder [37] defines realizers of maximal planar graphs in his study of the order dimension of planar graphs, and shows their application to planar straight line drawings [38] The construction of realizers of maximal planar graphs can also be efficiently parallelized [20] Brightwell and ....

M. Chrobak and S. Nakano. Minimum-width grid drawings of plane graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 104--110. Springer-Verlag, 1995.

.... to embed it, provided that method places vertices at distinct points in the plane (e.g. see [7, 16, 27] For example, if G is planar we can use any existing method for embedding G in the plane such that vertices are at grid points, and edges of the graph are straight lines that do not cross [5, 6, 10, 24, 25, 28]. Once the graph drawing is defined, we build a binary space partition tree on the vertices of this drawing. Each node v in this tree corresponds to a convex region R of the plane, and associated with v is a line that separates R into two regions, each of which are associated with a child of v. ....

M. Chrobak and S. Nakano. Minimum-width grid drawings of plane graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 104--110. Springer-Verlag, 1995.

....algorithm is a planar straight line drawing. The algorithm installs each v k , 4 = k = n, on a grid point, and each edge (v k ; w j ) p = j = q, does not intersect any edge of G k01 , as illustrated in Fig.5(c) Similar to the proof of Lemma 2 in [7] or the proof of Lemma 2 in [6], one can easily prove by induction on k that any number of executions of the shift operations for G k01 introduce no edge intersection in G k01 . Thus algorithm Shift Method outputs a planar straight line drawing of G. Next we estimate the area of a rectangular grid in which algorithm ....

....(except the location of vn = r y ) 6. Related topics In this section we survey related topics. Lower bounds For each n = 3, there exists a plane graph, as illustrated in Fig.13, which needs a rectangular grid with area at least b2(n0 1) 3c2 b2(n0 1) 3c for any planar straight line drawing [6]. Each nested triangle needs two more width and height. On the other hand we know a rectangular grid with area (n 0 2) 2 (n 0 2) is enough for planar straight line drawings. Thus there still exists a gap between upper and lower bounds for the area of the rectangular grid. The question what is ....

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M. Chrobak and S. Nakano, "Minimum-width grid drawings of plane graphs," Computational Geometry: Theory and Applications, 10, pp.29-54, 1998.

.... approach, Schnyder [34] gives an alternate lineartime scheme for producing an O(n) Theta O(n) integer grid drawing of a triangulated planar graph, whose edge resolution is O(n 3 ) Since then, several researchers have worked on extending and tightening these results in the integer grid model [5, 6, 29]. Several researchers have also considered trade offs involving the angular resolution (e.g. see [21, 22, 23] In addition, Di Battista, Tamassia, and Tollis [15] prove an interesting lower bound, which holds under any reasonable finite resolution rule, that producing a straight line ....

M. Chrobak and S. Nakano. Minimum-width grid drawings of plane graphs. Technical Report UCRCS -94-5, Dept. of Computer Sci., Univ. California, Riverside, 1994.

.... approach, Schnyder [44] gives an alternate linear time scheme for producing an O(n) Theta O(n) integer grid drawing of a triangulated planar graph, whose edge resolution is O(1=n) Since then, several researchers have worked on extending and tightening these results in the integer grid model [6, 7, 34]. We will refer to the method from [14, 15, 8] as the shift method, as it works by successively adding vertices to the drawing and shifting horizontally parts of the existing drawing. Several researchers have also considered tradeoffs involving the angular resolution (e.g. see [23, 24, 39] For ....

M. Chrobak and S. Nakano. Minimum-width grid drawings of plane graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 104--110. Springer-Verlag, 1995.

.... approach, Schnyder [44] gives an alternate linear time scheme for producing an O(n) Theta O(n) integer grid drawing of a triangulated planar graph, whose edge resolution is O(n 3 ) Since then, several researchers have worked on extending and tightening these results in the integer grid model [6, 7, 34]. We will refer to the method from [14, 15, 8] as the shift method, as it works by successively adding vertices to the drawing and shifting horizontally parts of the existing drawing. Several researchers have also considered trade offs involving the angular resolution (e.g. see [23, 24, 39] ....

M. Chrobak and S. Nakano. Minimum-width grid drawings of plane graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 104--110. Springer-Verlag, 1995.

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M. Chrobak, and S. Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29--54, 1998.

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Marek Chrobak and Shinichi Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29-54, 1998.

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