Drawing planar graphs with few segments on a polynomial grid

The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a drawing, one segment may represent multiple edges forming a path. Drawings of planar graphs with few segments were intensively studied in the past years. However, the area requirements were only considered for limited subclasses of planar graphs. In this paper, we show that trees have drawings with $3n/4-1$ segments and $n^2$ area, improving the previous result of $O(n^{3.58})$. We also show that 3-connected planar graphs and biconnected outerplanar graphs have a drawing with $8n/3-O(1)$ and $3n/2-O(1)$ segments, respectively, and $O(n^3)$ area.

[1]  Takao Nishizeki,et al.  Canonical Decomposition, Realizer, Schnyder Labeling And Orderly Spanning Trees Of Plane Graphs , 2005, Int. J. Found. Comput. Sci..

[2]  Antonios Symvonis,et al.  Simple Compact Monotone Tree Drawings , 2017, Graph Drawing.

[3]  Roberto Tamassia,et al.  Output-Sensitive Reporting of Disjoint Paths , 1996, Algorithmica.

[4]  Marc Noy,et al.  Augmenting the Connectivity of Outerplanar Graphs , 2008, Algorithmica.

[5]  Huaming Zhang,et al.  Canonical Ordering Trees and Their Applications in Graph Drawing , 2005, Discret. Comput. Geom..

[6]  Xin He,et al.  Compact Monotone Drawing of Trees , 2015, COCOON.

[7]  Alexander Wolff,et al.  Drawing Graphs on Few Circles and Few Spheres , 2017, CALDAM.

[8]  Dayu He,et al.  Nearly optimal monotone drawing of trees , 2016, Theor. Comput. Sci..

[9]  Md. Iqbal Hossain,et al.  Good spanning trees in graph drawing , 2015, Theor. Comput. Sci..

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

[11]  Yi-Ting Chiang,et al.  Orderly Spanning Trees with Applications , 2001, SIAM J. Comput..

[12]  Stephane Durocher,et al.  Drawing plane triangulations with few segments , 2018, Comput. Geom..

[13]  H. de Fraysseix,et al.  On topological aspects of orientations , 2001, Discret. Math..

[14]  Weidong Huang,et al.  A graph reading behavior: Geodesic-path tendency , 2009, 2009 IEEE Pacific Visualization Symposium.

[15]  Stefan Felsner,et al.  Schnyder Woods and Orthogonal Surfaces , 2008, Discret. Comput. Geom..

[16]  Stefan Felsner,et al.  Strongly Monotone Drawings of Planar Graphs , 2016, Symposium on Computational Geometry.

[17]  Stephane Durocher,et al.  A Note on Minimum-Segment Drawings of Planar Graphs , 2011, CCCG.

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

[19]  Alexander Wolff,et al.  On Monotone Drawings of Trees , 2014, Graph Drawing.

[20]  Martin Nöllenburg,et al.  On self-approaching and increasing-chord drawings of 3-connected planar graphs , 2016, J. Comput. Geom..

[21]  Nicolas Bonichon,et al.  Wagner's Theorem on Realizers , 2002, ICALP.

[22]  Wouter Meulemans,et al.  Experimental Analysis of the Accessibility of Drawings with Few Segments , 2017, GD.

[23]  André Schulz.  Drawing Graphs with Few Arcs , 2015, J. Graph Algorithms Appl..

[24]  Alexander Wolff,et al.  The Complexity of Drawing Graphs on Few Lines and Few Planes , 2016, WADS.

[25]  Wouter Meulemans,et al.  Drawing Planar Graphs with Few Geometric Primitives , 2017, WG.

[26]  G. Kant.  Algorithms for drawing planar graphs , 1993 .

[27]  Wouter Meulemans,et al.  Drawing Planar Cubic 3-Connected Graphs with Few Segments: Algorithms and Experiments , 2015, GD.

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

[29]  Xin He,et al.  Monotone Drawings of 3-Connected Plane Graphs , 2015, ESA.

[30]  Xin He,et al.  Optimal Monotone Drawings of Trees , 2016, SIAM J. Discret. Math..

[31]  Stefan Felsner,et al.  Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes , 2001, Order.

[32]  Stephen G. Kobourov,et al.  Gestalt Principles in Graph Drawing , 2015, GD.

[33]  Debajyoti Mondal,et al.  Minimum-segment convex drawings of 3-connected cubic plane graphs , 2010, J. Comb. Optim..

[34]  Peter Robinson,et al.  Drawability of Complete Graphs Using a Minimal Slope Set , 1994, Comput. J..