The Maximum Number of Edges in a Three-Dimensional Grid-Drawing

An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. The first universal lower bound on the volume of threedimensional grid-drawings is obtained as a corollary. Our results generalise to the setting of multi-dimensional polyline grid-drawings.

[1]  David R. Wood,et al.  Stacks, Queues and Tracks: Layouts of Graph Subdivisions , 2005, Discret. Math. Theor. Comput. Sci..

[2]  Emilio Di Giacomo,et al.  Track Drawings of Graphs with Constant Queue Number , 2003, GD.

[3]  János Pach,et al.  Three-dimensional Grid Drawings of Graphs , 1997, GD.

[4]  David R. Wood,et al.  Layouts of Graph Subdivisions , 2004, GD.

[5]  Toru Hasunuma,et al.  Laying Out Iterated Line Digraphs Using Queues , 2003, GD.

[6]  David R. Wood,et al.  Three-Dimensional Grid Drawings with Sub-quadratic Volume , 2003, GD.

[7]  David R. Wood,et al.  Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing , 2002, FSTTCS.

[8]  J. Pach.  Towards a Theory of Geometric Graphs , 2004 .

[9]  Andrea Sterbini,et al.  3D Straight-Line Grid Drawing of 4-Colorable Graphs , 1997, Inf. Process. Lett..

[10]  Pat Morin,et al.  Path-Width and Three-Dimensional Straight-Line Grid Drawings of Graphs , 2002, GD.

[11]  Stephen K. Wismath,et al.  Drawing Kn in Three Dimensions with Two Bends per Edge , 2004 .

[12]  David R. Wood,et al.  Tree-Partitions of k-Trees with Applications in Graph Layout , 2003, WG.

[13]  Pat Morin,et al.  Three-Dimensional 1-Bend Graph Drawings , 2004, J. Graph Algorithms Appl..

[14]  Stefan Felsner,et al.  Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions , 2001, J. Graph Algorithms Appl..

[15]  Pat Morin,et al.  Layout of Graphs with Bounded Tree-Width , 2004, SIAM J. Comput..

[16]  Emilio Di Giacomo,et al.  Drawing Series-Parallel Graphs on Restricted Integer 3D Grids , 2003, GD.

[17]  Emilio Di Giacomo,et al.  Drawing series-parallel graphs on a box , 2002, CCCG.