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 n-vertex 3-connected planar graphs: ● 2D convex grid drawings with (3n) x (3n/2) area under the edge L1-resolution rule; ● 2D strictly convex grid drawings with 0(n3) x O(T13 ) area under the edge resolution rule; c 2D strictly convex drawings with O(1) x O(n) area under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers; . 3D convex drawings with O(1) x O(1) x O(n) volume under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers, We also show the following lower bounds: ● For infinitely many n-vertex graphs G, if G has a straightline 2D convex drawing in a w x h gnd satisfying the edge L1-resolution rule then w, h > 5n/6 + Q(l) a,.d w + h ~ 8n/3 + 0(1). ● For infinitely many bounded-degree trlconnected planar graphs G with n vertices, any 3D convex drawing of G must have volume 2QfnJ under the angular resolution rule,

[1]  Steven Skiena,et al.  Complexity aspects of visibility graphs , 1995, Int. J. Comput. Geom. Appl..

[2]  Marek Chrobak,et al.  A Linear-Time Algorithm for Drawing a Planar Graph on a Grid , 1995, Inf. Process. Lett..

[3]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[4]  D. Rose,et al.  Generalized nested dissection , 1977 .

[5]  Gerhard J. Woeginger,et al.  Drawing graphs in the plane with high resolution , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[6]  János Pach,et al.  Small sets supporting fary embeddings of planar graphs , 1988, STOC '88.

[7]  Bernd Becker,et al.  Layouts with Wires of Balanced Length , 1987, Inf. Comput..

[8]  Giuseppe Liotta,et al.  Computing Proximity Drawings of Trees in the 3-Dimemsional Space , 1995, WADS.

[9]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[10]  Michael B. Dillencourt,et al.  A linear-time algorithm for testing the inscribability of trivalent polyhedra , 1992, SCG '92.

[11]  Peter Eades,et al.  The Techniques of Komolgorov and Bardzin for Three-Dimensional Orthogonal Graph Drawings , 1996, Inf. Process. Lett..

[12]  Steven P. Reiss.  An Engine for the 3D Visualization of Program Information , 1995, J. Vis. Lang. Comput..

[13]  Bernd Becker,et al.  Layouts with Wires of Balanced Length , 1985, STACS.

[14]  W. T. Tutte.  Convex Representations of Graphs , 1960 .

[15]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[16]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[17]  Carsten Thomassen,et al.  Planarity and duality of finite and infinite graphs , 1980, J. Comb. Theory B.

[18]  R. Tarjan,et al.  The analysis of a nested dissection algorithm , 1987 .

[19]  Ioannis G. Tollis,et al.  Area requirement and symmetry display of planar upward drawings , 1992, Discret. Comput. Geom..

[20]  W. Whiteley,et al.  Statics of Frameworks and Motions of Panel Structures: A projective Geometric Introduction , 1982 .

[21]  Steven P. Reiss.  3-D Visualization of Program Information , 1994, Graph Drawing.

[22]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[23]  Goos Kant,et al.  Drawing planar graphs using the lmc-ordering , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[24]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[25]  Peter Eades,et al.  Drawing Stressed Planar Graphs in Three Dimensions , 1995, GD.

[26]  R. Connelly.  Rigidity and energy , 1982 .

[27]  Tao Lin,et al.  Three-Dimensional Graph Drawing , 1994, Graph Drawing.

[28]  Seth M. Malitz,et al.  On the angular resolution of planar graphs , 1992, STOC '92.

[29]  Marek Chrobak,et al.  Minimum-width grid drawings of plane graphs , 1994, Comput. Geom..

[30]  S. Mehdi Hashemi,et al.  Upward Drawings to Fit Surfaces , 1994, ORDAL.

[31]  Ioannis G. Tollis,et al.  Algorithms for Drawing Graphs: an Annotated Bibliography , 1988, Comput. Geom..

[32]  Roberto Tamassia,et al.  On Line Convex Planarity Testing , 1994 .

[33]  Bernd Becker,et al.  On the Optimal Layout of Planar Graphs with Fixed Boundary , 1987, SIAM J. Comput..

[34]  George E. Andrews,et al.  A LOWER BOUND FOR THE VOLUME OF STRICTLY CONVEX BODIES WITH MANY BOUNDARY LATTICE POINTS , 1963 .

[35]  Roberto Tamassia,et al.  On-Line Convex Plabarity Testing , 1994, WG.

[36]  Michael T. Goodrich,et al.  On the Complexity of Approximating and Illuminating Three-Dimensional Convex Polyhedra (Preliminary Version) , 1995, WADS.

[37]  W. Whiteley.  Motions and stresses of projected polyhedra , 1982 .

[38]  M. Chrobak,et al.  Convex Grid Drawings of 3-Connected Planar Graphs , 1997, Int. J. Comput. Geom. Appl..

[39]  Roberto Tamassia,et al.  Planar Drawings and Angular Resolution: Algorithms and Bounds (Extended Abstract) , 1994, ESA.

[40]  Thierry Jéron,et al.  3D Layout of Reachability Graphs of Communicating Processes , 1994, Graph Drawing.

[41]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

[42]  Warren D. Smith,et al.  A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere , 1992, math/9210218.

[43]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[44]  H. S. M. Coxeter,et al.  Vorlesungen über die Theorie der Polyeder , 1935 .

[45]  János Pach,et al.  How to draw a planar graph on a grid , 1990, Comb..