S. Felsner, G. Liotta, and S. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions. In Proc. Graph Drawing, GD 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for short layered drawings. The proof is similar to the proof of Lemma 15 for short layered drawings. We simply change all references to short layered drawings to unconstrained layered drawings and all references to Lemma 14 to Lemma 29. This bound is also proved independently by Felsner et al. in [9]. Lemma 30 For every graph G, any unconstrained layered drawing of G occupies at least pw(G) layers. Short layered drawings are unconstrained layered drawings so, by Lemma 16, our lower bound is optimal. Corollary 31 For each h 1, there exists a graph G with pw(G) h and an unconstrained ....

Stefan Felsner, Giuseppe Liotta, and Stephen K. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions, manuscript. http://page.inf.fu-berlin.de/~felsner/Paper/rest-grid.ps.gz, 2002.

....upper and lower bounds for each type of layered drawing (proper, short, upright, unconstrained) and give linear time algorithms for obtaining drawings matching each upper bound. We note that the optimality of the upper bound for unconstrained layered drawings contradicts Proposition 1 of [8], and the optimality of the upper bound for short layered drawings contradicts Theorem 2 also of [8] 1 Introduction An h layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its ....

....and give linear time algorithms for obtaining drawings matching each upper bound. We note that the optimality of the upper bound for unconstrained layered drawings contradicts Proposition 1 of [8] and the optimality of the upper bound for short layered drawings contradicts Theorem 2 also of [8]. 1 Introduction An h layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, at if ....

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Stefan Felsner, Giuseppe Liotta, and Stephen K. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions (extended abstract). In Petra Mutzel, Michael Junger, and Sebastian Leipert, editors, Graph Drawing, 9th International Symposium (GD 2001.

....for short layered drawings. The proof is similar to the proof of Lemma 15 for short layered drawings. We simply change all references to short layered drawings to unconstrained layered drawings and all references to Lemma 14 to Lemma 29. This bound is also proved independently by Felsner et al. in [9]. Lemma 30 For every graph G, any unconstrained layered drawing of G occupies at least pw(G) layers. Short layered drawings are unconstrained layered drawings so, by Lemma 16, our lower bound is optimal. h and an unconstrained h layer drawing. in unconstrained layered drawings of trees are ....

Stefan Felsner, Giuseppe Liotta, and Stephen K. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions, manuscript. http://page.inf. fu-berlin.de/~felsner/Paper/rest-grid.ps.gz, 2002. 17

....upper and lower bounds for each type of layered drawing (proper, short, upright, unconstrained) and give linear time algorithms for obtaining drawings matching each upper bound. We note that the optimality of the upper bound for unconstrained layered drawings contradicts Proposition 1 of [8], and the optimality of the upper bound for short layered drawings contradicts Theorem 2 also of [8] 1 Introduction An h layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its ....

....and give linear time algorithms for obtaining drawings matching each upper bound. We note that the optimality of the upper bound for unconstrained layered drawings contradicts Proposition 1 of [8] and the optimality of the upper bound for short layered drawings contradicts Theorem 2 also of [8]. 1 Introduction An h layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat ....

[Article contains additional citation context not shown here]

Stefan Felsner, Giuseppe Liotta, and Stephen K. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions (extended abstract). In Petra Mutzel, Michael Junger, and Sebastian Leipert, editors, Graph Drawing, 9th International Symposium (GD 2001.

....Z. That is, the volume of a polyline drawing is the number of gridpoints in the bounding box. This definition is formulated so that two dimensional drawings have positive volume. We are interested in polyline drawings with small volume. The volume of straight line drawings has been widely studied [21, 26, 31, 37, 41, 42, 51, 103, 107, 125]. Three dimensional graph drawings in which the vertices are allowed real coordinates have also been studied [23, 28, 29, 34, 43, 59, 78 81, 93, 102] Aesthetic criteria besides volume that have been considered include symmetry [78 81] aspect ratio [29, 59] angular resolution [29, 59] ....

....polyline drawings, including those established in this paper. Table 2: Volume of polyline drawings of graphs with n vertices and m n edges. graph family bends per edge volume reference ) 31] 103] K h minor free 0 genus # 0 planar 0 outerplanar 0 [51] constant tree width 0 [42] c colourable q queue 1 O(cqm) q queue 2 O(qn) q queue (constant # 0) O(1) O(mq q queue q) log q) Theorem 24 Cohen et al. 31] proved that every graph has a straight line drawing with ) volume, and that this bound is asymptotically optimal ....

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S. FELSNER, G. LIOTTA, AND S. WISMATH, Straight-line drawings on restricted integer grids in two and three dimensions. In [98], pp. 328--342.

....Z drawing with volume X Z. That is, the volume of a 3D drawing is the number of gridpoints in the bounding box. This definition is formulated so that 2D drawings have positive volume. We are interested in 3D drawings with small volume. The volume of 3D drawings has been widely studied [3, 6, 9, 11, 14, 16, 19, 36 38, 40]. Three dimensional graph drawings in which the vertices are allowed real coordinates have also been studied [5, 7, 8, 10, 17, 21, 26 29, 32, 35] Aesthetic criteria besides volume which have been considered include symmetry [26 29] aspect # Research supported by NSERC and FCAR. Completed ....

....Upper bounds on the volume of 3D drawings of graphs with n vertices and m edges. graph family volume reference ) Cohen et al. 9] maximum degree # Theorem 3 constant maximum degree ) Pach et al. 36, 37] no K h minor ( h constant) constant genus outerplanar Felsner et al. [19] constant tree width Dujmovi c and Wood [16] Cohen et al. 9] proved that every graph has a 3D drawing with ) volume, and that this bound is asymptotically optimal for complete graphs K n . Our edge sensitive bounds of are greater than ) in the worst case. It is an open problem ....

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S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In [33], pp. 328--342.

....grid drawings with linear volume, which is the largest known class of graphs admitting such drawings. Motivated by applications in information visualisation, VLSI layout, and software engineering (see [12] there is a growing body of research in three dimensional straight line graph drawing [4, 6, 8, 12, 13, 24, 26, 35]. The remainder of the paper is organised as follows. Section 1.1 recalls a number of definitions and well known results. In Sections 1.2, 1.3 and 1.4 we survey and state our results for tree partitions, queue layouts and threedimensional graph drawings, respectively. In Section 2 we prove the ....

....a three dimensional drawing is contained in an axis aligned box with side lengths X 1, Y 1 and Z 1, then we speak of a X Y Z drawing with volume X Y Z. We are interested in three dimensional drawings with small volume. The volume of three dimensional drawings has been extensively studied [4, 6, 8, 12, 13, 24, 26, 35]. Cohen, Eades, Lin, and Ruskey [6] proved that every graph has a three dimensional drawing with O(n ) volume, and this bound is asymptotically tight for the complete graph Kn . It is therefore of interest to identify fixed graph parameters which allow for three dimensional drawings with O(n ....

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S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on pp. 328--342, Springer, 2002.

....the complete bipartite graph with equal sized bipartitions. If p is a suitably chosen prime, the main step of this algorithm represents the vertices in the ith colour class by grid points in the set f(i; t; it) t i (mod p)g : The first linear volume bound was established by Felsner et al. [17], who proved that every outerplanar graph has a drawing with O(n) volume. Their elegant algorithm wraps a two dimensional layered drawing around a triangular prism; see Lemma 5 for more on this method. Poranen [32] proved that seriesparallel digraphs have upward three dimensional drawings with ....

....For example, outerplanar graphs, series parallel graphs and Halin graphs respectively have tree width 2, 2 and 3 (see [2, 12] Thus Corollary 1(b) implies that these graphs have three dimensional drawings with O(n log n) volume. While linear volume is possible for outerplanar graphs [17], our result is the first known sub quadratic volume bound for all series parallel and Halin graphs. Another example arises in software engineering applications. Thorup [34] proved that the control flow graphs of go to free programs in many programming languages have tree width bounded by a small ....

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S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In [28], pp. 328--342.

....planar embedding [3] Some planar graphs clearly can be drawn with less than O(n ) area. An interesting question is therefore whether such graphs can also be drawn with less than O(n ) area and with a constant aspect ratio. In particular, Steve Wismath at the 2001 Graph Drawing Symposium [2] conjectured that this is not possible for a graph containing K 2;n , which can clearly be drawn in O(n) area. In this note, we show that this is indeed the case: no drawing of K 2;n with O(n) area has constant aspect ratio. 2 Preliminaries K 2;n is the complete bipartite graph with two vertices ....

S. Felsner, G. Liotta, and S. Wismath. Straightline drawings on restricted integer grids in two and three dimensions. In Proc. Graph Drawing (GD 2001.

....proved that every k colourable graph, for fixed k 2, has a three dimensional drawing with O(n ) volume, and that this bound is asymptotically optimal for the complete bipartite graph with equal sized bipartitions. The first linear volume bound was established by Felsner, Wismath, and Liotta [14], who proved that every outerplanar graph has a drawing with O(n) volume. Poranen [25] proved that series parallel digraphs have upward three dimensional drawings with O(n ) volume, and that this bound can be improved to O(n ) and O(n) in certain special cases. di Giacomo, Liotta, and ....

....et al. 12] every planar graph has a three dimensional drawing with O(n ) volume. This result also follows from the classical algorithms of de Fraysseix et al. 6] and Schnyder [27] for producing plane grid drawings. This paper is motivated by the following open problem due to Felsner et al. [14]. Open Problem 1 ( 14] Does every planar graph have a three dimensional drawing with O(n) volume In fact, any o(n ) bound would be of interest. In this paper we prove that Open Problem 1 is almost equivalent to an existing open problem in the theory of queue layouts. 1.3 Queue layouts ....

[Article contains additional citation context not shown here]

S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328--342, Springer, 2002.

....the complete bipartite graph with equal sized bipartitions. If p is a suitably chosen prime, the main step of this algorithm represents the vertices in the ith colour class by grid points in the set f(i; t; it) t i (mod p)g : The first linear volume bound was established by Felsner et al. [15], who proved that every outerplanar graph has a drawing with O(n) volume. Their elegant algorithm wraps a two dimensional layered drawing around a triangular prism; see Lemma 4 for more on this method. Poranen [29] proved that seriesparallel digraphs have upward three dimensional drawings with ....

....treewidth. For example, outerplanar graphs, series parallel graphs and Halin graphs respectively have treewidth 2, 2 and 3 (see [2, 11] Thus Corollary 1(b) implies that these graphs have three dimensional drawings with O(n log n) volume. While linear volume is possible for outerplanar graphs [15], our result is the first known sub quadratic volume bound for all series parallel and Halin graphs. Another example arises in software engineering applications. Thorup [30] proved that the control flow graphs of go to free programs in many programming languages have treewidth bounded by a small ....

[Article contains additional citation context not shown here]

S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In [25], pp. 328--342.

....proved that every k colourable graph, for fixed k 2, has a three dimensional drawing with O(n ) volume, and that this bound is asymptotically optimal for the complete bipartite graph with equal sized bipartitions. The first linear volume bound was established by Felsner, Wismath, and Liotta [11], who proved that every outerplanar graph has a drawing with O(n) volume. Poranen [21] proved that series parallel digraphs have upward threedimensional drawings with O(n ) volume, and that this bound can be improved to O(n ) and O(n) in certain special cases. di Giacomo, Liotta, and Wismath ....

....et al. 10] every planar graph has a three dimensional drawing with O(n ) volume. This result also follows from the classical algorithms of de Fraysseix et al. 5] and Schnyder [23] for producing plane grid drawings. This paper is motivated by the following open problem due to Felsner et al. [11]. Open Problem 1 ( 11] Does every planar graph have a three dimensional drawing with O(n) volume In fact, any o(n ) bound would be of interest. In this paper we reduce this question to an open problem in the theory of queue layouts. 1.3 Queue layouts For a graph G, a linear order of V ....

[Article contains additional citation context not shown here]

S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328--342, Springer, 2002.

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S. Felsner, G.Liotta, and S. Wismath. Straight line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. Junger, and S. Leipert, editors, Graph Drawing (Proc. GD '01), volume 2265 of Lecture Notes Comput. Sci. Springer-Verlag, 2001.

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S. Felsner, G. Liotta and S. Wismath. Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions. In M. Junger and P. Mutzel, editor, Graph Drawing (Proc. GD '01), volume 2265 of Lecture Notes Comput. Sci., pages 328--343. Springer-Verlag, 2002.

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S. Felsner, G.Liotta, and S. Wismath. Straight line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. Junger, and S. Leipert, editors, Graph Drawing (Proc. GD '01), volume 2265 of Lecture Notes Comput. Sci. Springer-Verlag, 2001.

....to the notion of path width, tree width and queue number. 3.1 Restricted grids One approach to solving the 3D case for planar graphs is to choose a xed shape grid (of linear volume) and ask what class of graphs can be drawn on that grid. Fig. 7. A box and a prism Felsner, Liotta and Wismath [15] studied two di erent types of n 2 2 grids . A box is a n 2 2 grid where each side of the bounding box is also a typically 3D grids are measured by the number of lines in each dimension grid line. Therefore, a box has four tracks which lie on two parallel planes and are one grid unit ....

....8. A prism drawable graph G and its drawing planar graphs. For example, Figure 8 shows a maximal planar graph, and its prism drawing. The rst non trivial class of graphs drawable in linear volume are the outerplanar graphs . v v Fig. 9. An outerplanar graph drawn by Step 1 . Theorem 5. [15] All outerplanar graphs can be drawn on a prism. outerplanar means 9 a plane embedding with all vertices on the exterior face Proof. Let G be an outerplanar graph with a speci ed outerplanar embedding. First a 2D drawing of G is computed on a grid that consists of O(n) horizontal tracks and ....

S. Felsner, S. Wismath, and G. Liotta. Straight-line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. Junger, and S. Leipert, editors, Proc. 9th International Symp. on Graph Drawing (GD '01), volume 2265 of Lecture Notes in Comput. Sci., pages 328-342. Springer, 2002.

....a much more powerful grid than the prism, we prove that not all planar graphs are box drawable. Several recent related results about 3D straight line drawings of limited volume have been published after the conference version of this paper was presented at the Symposium on Graph Drawing GD 2001 [22]. Dujmovic, Morin, and Wood [20] present O(n log n) volume drawings of graphs with bounded tree width and O(n) volume for graphs with bounded path width. Wood [42] shows that also graphs with bounded queue number have 3D straight line grid drawings of O(n) volume. A very recent result by ....

....not drawable on tracks. Furthermore any graph containing K 4 as a subgraph is not track drawable. Given a drawing of a graph on an n k grid (for k 3) we can attach a copy of K 4 which makes the resulting graph not track drawable. In the extended abstract for the graph drawing conference GD 01[22] we incorrectly argued that for trees n k grid drawable is equivalent to k track drawable. It was rst observed by Matthew Suderman that this was false. In a recent manuscript Suderman [40] describes a family S of trees, such that S can be drawn on the n (k 1) grid but requires 2k 1 ....

S. Felsner, G. Liotta and S. Wismath. Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions. In M. Junger and P. Mutzel, editor, Graph Drawing (Proc. GD '01), volume 2265 of Lecture Notes Comput. Sci., pages 328-343. Springer-Verlag, 2002.

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S. Felsner, G. Liotta, and S. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions. In Proc. Graph Drawing, GD 2001.

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S. FELSNER, G. LIOTTA, AND S. WISMATH, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328--342, Springer, 2002.

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S. FELSNER, G. LIOTTA, AND S. WISMATH, Straight-line drawings on restricted integer grids in two and three dimensions. In [75], pp. 328--342.

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S. Felsner, G. Liotta, and S. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions, J. Graph Algorithms Appl., 7 (2003), pp. 363-398.

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S. Felsner, G. Liotta, and S. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. J unger, and S. Leipert, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328-342, Springer, 2002.

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S. FELSNER, G. LIOTTA, AND S. WISMATH, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328--342, Springer, 2002.

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Stefan Felsner, Giussepe Liotta, and Stephen Wismath, Straight-line drawings on restricted integer grids in two and three dimensions, Proc. 9th International Symp. on Graph Drawing (GD '01) (Petra Mutzel, Michael Junger, and Sebastian Leipert, eds.), Lecture Notes in Comput. Sci., vol. 2265, Springer, 2002, pp. 328-342.

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S. Felsner, G. Liotta, and S. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions. In Proc. Graph Drawing, GD 2001.

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S. FELSNER, G. LIOTTA, AND S. WISMATH, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328--342, Springer, 2002.

No context found.

S. Felsner, G. Liotta, and S. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. J unger, and S. Leipert, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328-342, Springer, 2002.

No context found.

S. Felsner, S. Wismath, and G. Liotta, Straight-line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. J unger, and S. Leipert, eds., Proc. 9th pp. 328-342, Springer, 2002.

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S. FELSNER, S. WISMATH, AND G. LIOTTA, Straight-line drawings on restricted integer grids in two and three dimensions. In P. MUTZEL, M. J UNGER, AND S. LEIPERT, eds., Proc. 9th International Symp. on Graph Drawing (GD '01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328--342, Springer, 2002.

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