Drawing Subcubic 1-Planar Graphs with Few Bends, Few Slopes, and Large Angles

We show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each. This bound is obtained by drawings whose vertex and crossing resolution is at least \(\pi /4\). On the other hand, if the embedding is fixed, then there is a 3-connected cubic 1-planar graph that needs 3 slopes when drawn with at most 1 bend per edge. We also show that 2 slopes always suffice for 1-planar drawings of subcubic 1-planar graphs with at most 2 bends per edge. This bound is obtained with vertex resolution \(\pi /2\) and the drawing is RAC (crossing resolution \(\pi /2\)). Finally, we prove lower bounds for the slope number of straight-line 1-planar drawings in terms of number of vertices and maximum degree.

[1]  Yanpei Liu,et al.  A Linear Algorithm for 2-bend Embeddings of Planar Graphs in the Two-dimensional Grid , 1998, Discret. Appl. Math..

[2]  Stephen G. Kobourov,et al.  Polar Coordinate Drawing of Planar Graphs with Good Angular Resolution , 2001, Graph Drawing.

[3]  Emilio Di Giacomo,et al.  Ortho-polygon Visibility Representations of Embedded Graphs , 2016, Algorithmica.

[4]  Jan Kratochvíl,et al.  The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree , 2013, Graphs Comb..

[5]  Goos Kant.  Hexagonal Grid Drawings , 1992, WG.

[6]  Steven Chaplick,et al.  1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area , 2009 .

[7]  Franz-Josef Brandenburg,et al.  T-shape visibility representations of 1-planar graphs , 2017, Comput. Geom..

[8]  Giuseppe Liotta,et al.  Right angle crossing graphs and 1-planarity , 2013, Discret. Appl. Math..

[9]  Michael A. Bekos,et al.  On RAC drawings of 1-planar graphs , 2017, Theor. Comput. Sci..

[10]  Giuseppe Liotta,et al.  Planar and Plane Slope Number of Partial 2-Trees , 2013, Graph Drawing.

[11]  Walter Didimo,et al.  A Survey on Graph Drawing Beyond Planarity , 2018, ACM Comput. Surv..

[12]  Goos Kant,et al.  Drawing planar graphs using the canonical ordering , 1996, Algorithmica.

[13]  Emilio Di Giacomo,et al.  Drawing Outer 1-planar Graphs with Few Slopes , 2014, Graph Drawing.

[14]  Michael A. Bekos,et al.  Planar Octilinear Drawings with One Bend Per Edge , 2014, Graph Drawing.

[15]  Piotr Micek,et al.  Outerplanar graph drawings with few slopes , 2014, Comput. Geom..

[16]  Tomás Madaras,et al.  The structure of 1-planar graphs , 2007, Discret. Math..

[17]  Roberto Tamassia,et al.  Algorithms for Plane Representations of Acyclic Digraphs , 1988, Theor. Comput. Sci..

[18]  Jirí Matousek,et al.  Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness , 2006, Electron. J. Comb..

[19]  János Pach,et al.  Bounded-Degree Graphs can have Arbitrarily Large Slope Numbers , 2006, Electron. J. Comb..

[20]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[21]  Michael A. Bekos,et al.  A Universal Slope Set for 1-Bend Planar Drawings , 2017, Symposium on Computational Geometry.

[22]  G. Ringel.  Ein Sechsfarbenproblem auf der Kugel , 1965 .

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

[24]  Giuseppe Liotta,et al.  An annotated bibliography on 1-planarity , 2017, Comput. Sci. Rev..

[25]  Michael T. Goodrich,et al.  Planar Orthogonal and Polyline Drawing Algorithms , 2013, Handbook of Graph Drawing and Visualization.

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

[27]  Dömötör Pálvölgyi,et al.  Drawing cubic graphs with the four basic slopes , 2011, Graph Drawing.

[28]  David Eppstein,et al.  Drawings of planar graphs with few slopes and segments , 2007, Comput. Geom..

[29]  Emilio Di Giacomo,et al.  Drawing subcubic planar graphs with four slopes and optimal angular resolution , 2018, Theor. Comput. Sci..

[30]  Giuseppe Liotta,et al.  Fáry's Theorem for 1-Planar Graphs , 2012, COCOON.

[31]  Goos Kant,et al.  A Better Heuristic for Orthogonal Graph Drawings , 1994, ESA.

[32]  Balázs Keszegh,et al.  Drawing Planar Graphs of Bounded Degree with Few Slopes , 2010, Graph Drawing.

[33]  Stephen G. Kobourov,et al.  Straight-Line Grid Drawings of 3-Connected 1-Planar Graphs , 2013, Graph Drawing.