Really Straight Graph Drawings

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes, and that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of non-planar graphs with few slopes are also considered. For example, it is proved that graphs of bounded degree and bounded treewidth have drawings with $\mathcal{O}({\rm log} n)$ slopes.

[1]  Ralph Freese,et al.  Automated Lattice Drawing , 2004, ICFCA.

[2]  Jurek Czyzowicz.  Lattice diagrams with few slopes , 1991, J. Comb. Theory, Ser. A.

[3]  Rudolf Halin,et al.  Tree-partitions of infinite graphs , 1991, Discret. Math..

[4]  Petr A. Golovach,et al.  Interval degree and bandwidth of a graph , 2003, Discret. Appl. Math..

[5]  Shin-Ichi Nakano,et al.  Rectangular drawings of plane graphs without designated corners , 2002, Comput. Geom..

[6]  David R. Wood,et al.  Characterisations of intersection graphs by vertex orderings , 2004, Australas. J Comb..

[7]  Md. Saidur Rahman,et al.  Rectangular drawings of planar graphs , 2002, J. Algorithms.

[8]  Micha Sharir,et al.  A tight bound for the number of different directions in three dimensions , 2003, SCG '03.

[9]  Detlef Seese,et al.  Tree-partite graphs and the complexity of algorithms , 1985, FCT.

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

[11]  Zhen Zhang.  A note on arrays of dots with distinct slopes , 1993, Comb..

[12]  Jeremy L. Martin.  On the topology of graph picture spaces , 2003 .

[13]  David R. Wood,et al.  On Linear Layouts of Graphs , 2004, Discret. Math. Theor. Comput. Sci..

[14]  Jeremy L. Martin.  THE SLOPES DETERMINED BY n POINTS IN THE PLANE , 2003 .

[15]  Jeremy L. Martin.  Geometry of graph varieties , 2003, math/0302089.

[16]  Paul Erdös,et al.  Bounds for arrays of dots with distinct slopes or lengths , 1992, Comb..

[17]  Shin-Ichi Nakano,et al.  Rectangular grid drawings of plane graphs , 1998, Comput. Geom..

[18]  G. Dirac.  On rigid circuit graphs , 1961 .

[19]  David Eppstein,et al.  Separating Thickness from Geometric Thickness , 2002, GD.

[20]  S. M. Malitz.  Graphs with E edges have pagenumber O(√E) , 1994, FOCS 1994.

[21]  Prosenjit Bose,et al.  Drawing Nice Projections of Objects in Space , 1999, J. Vis. Commun. Image Represent..

[22]  Robert E. Jamison,et al.  Few slopes without collinearity , 1986, Discret. Math..

[23]  Peter Ungar,et al.  2N Noncollinear Points Determine at Least 2N Directions , 1982, J. Comb. Theory, Ser. A.

[24]  Robert E. Jamison,et al.  Planar configurations which determine few slopes , 1984 .

[25]  Sartaj Sahni,et al.  A linear algorithm to find a rectangular dual of a planar triangulated graph , 1986, 23rd ACM/IEEE Design Automation Conference.

[26]  Bogdan Oporowski,et al.  On tree-partitions of graphs , 1996, Discret. Math..

[27]  Andrzej Pelc,et al.  Drawing orders with few slopes , 1990, Discret. Math..

[28]  Greg A. Wade,et al.  Drawability of Complete Graphs Using a Minimal Slope Set , 1994, Comput. J..

[29]  Sartaj Sahni,et al.  A linear time algorithm to check for the existence of a rectangular dual of a planar triangulated graph , 1987, Networks.

[30]  David Eppstein,et al.  The geometric thickness of low degree graphs , 2003, SCG '04.

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

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

[33]  Peter Eades,et al.  Finding the Best Viewpoints for Three-Dimensional Graph Drawings , 1997, GD.

[34]  Robert E. Jamison.  Direction trees , 1987, Discret. Comput. Geom..

[35]  R. E Peile,et al.  Sets of points with pairwise distinct slopes , 2000 .

[36]  Andrzej Pelc,et al.  Crooked diagrams with few slopes , 1990 .

[37]  P. Scott.  On the Sets of Directions Determined by n Points , 1970 .

[38]  Bogdan Oporowski,et al.  Some results on tree decomposition of graphs , 1995, J. Graph Theory.

[39]  Peter Ungar.  On Diagrams Representing Maps , 1953 .

[40]  Micha Sharir,et al.  Solution of Scott's problem on the number of directions determined by a point set in 3-space , 2004, SCG '04.

[41]  Seth M. Malitz,et al.  Graphs with E Edges Have Pagenumber O(sqrt(E)) , 1994, J. Algorithms.

[42]  Robert E. Jamison.  Structure of slope-critical configurations , 1984 .

[43]  Shin-Ichi Nakano,et al.  Box-Rectangular Drawings of Plane Graphs , 1999, J. Algorithms.

[44]  Michael E. Houle,et al.  Approximation Algorithms for Finding Best Viewpoints , 1998, GD.

[45]  F. Leighton,et al.  Drawing Planar Graphs Using the Canonical Ordering , 1996 .

[46]  Matthew Suderman,et al.  Pathwidth And Layered Drawings Of Trees , 2004, Int. J. Comput. Geom. Appl..

[47]  Joost Engelfriet,et al.  Domino Treewidth , 1997, J. Algorithms.

[48]  P. R. Scott.  On the Sets of Directions Determined by n Points , 1970 .

[49]  Rom Pinchasi,et al.  A note on the minimum number of edge-directions of a convex polytope , 2004, J. Comb. Theory, Ser. A.

[50]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[51]  David Eppstein,et al.  Journal of Graph Algorithms and Applications Geometric Thickness of Complete Graphs , 2022 .

[52]  Roberto Tamassia,et al.  On the Computational Complexity of Upward and Rectilinear Planarity Testing , 1994, SIAM J. Comput..