On 3D visibility representations of graphs with few crossings per edge

Abstract A graph is k-planar if it can be drawn in the plane such that each edge is crossed at most k > 0 times. These graphs represent a natural extension of planar graphs and they are among the most investigated families in the growing field of graph drawing beyond planarity. In this paper, we study visibility representations of k-planar graphs in three dimensions. In particular, we provide a technique for a meaningful family of 2-planar graphs, called 5-kite-augmented graphs, which include as subgraphs all 1-planar graphs and all simple optimal 2-planar graphs (i.e., those 2-planar graphs that attain the maximum number of edges). We prove that every 5-kite-augmented graph has a z-parallel visibility representation, i.e., a three-dimensional visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of the input graph, which is a well-known type of visibility representation in two dimensions.

[1]  Michael A. Bekos,et al.  3D Visibility Representations of 1-planar Graphs , 2017, GD.

[2]  Joan P. Hutchinson,et al.  Rectangle-visibility Representations of Bipartite Graphs , 1994, Discret. Appl. Math..

[3]  Jan Stola.  Unimaximal Sequences of Pairs in Rectangle Visibility Drawing , 2008, Graph Drawing.

[4]  Alexander Grigoriev,et al.  Algorithms for Graphs Embeddable with Few Crossings per Edge , 2005, Algorithmica.

[5]  Thomas C. Shermer,et al.  On representations of some thickness-two graphs , 1999, Comput. Geom..

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

[7]  Michael A. Bekos,et al.  On Optimal 2- and 3-Planar Graphs , 2017, Symposium on Computational Geometry.

[8]  Giuseppe Liotta,et al.  Embedding-Preserving Rectangle Visibility Representations of Nonplanar Graphs , 2018, Discret. Comput. Geom..

[9]  János Pach,et al.  Graphs drawn with few crossings per edge , 1997, Comb..

[10]  Giuseppe Liotta,et al.  Ortho-polygon Visibility Representations of 3-connected 1-plane Graphs , 2018, GD.

[11]  Michael Kaufmann,et al.  Bar 1-Visibility Graphs vs. other Nearly Planar Graphs , 2014, J. Graph Algorithms Appl..

[12]  Stephen K. Wismath,et al.  Characterizing bar line-of-sight graphs , 1985, SCG '85.

[13]  Sándor P. Fekete,et al.  Rectangle and Box Visibility Graphs in 3D , 1999, Int. J. Comput. Geom. Appl..

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

[15]  Michael A. Bekos,et al.  Edge partitions of optimal 2-plane and 3-plane graphs , 2019, Discret. Math..

[16]  M. Kaufmann.  Bar 1-Visibility Graphs vs. other Nearly Planar Graphs , 2014 .

[17]  Sándor P. Fekete,et al.  New Results on a Visibility Representation of Graphs in 3D , 1995, GD.

[18]  Michael A. Bekos,et al.  Guest Editors' Foreword and Overview , 2018, J. Graph Algorithms Appl..

[19]  Hazel Everett,et al.  A Visibility Representation for Graphs in Three Dimensions , 1998, J. Graph Algorithms Appl..

[20]  Vladimir P. Korzhik,et al.  Minimal Obstructions for 1‐Immersions and Hardness of 1‐Planarity Testing , 2013, J. Graph Theory.

[21]  Thomas C. Shermer,et al.  On Rectangle Visibility Graphs. III. External Visibility and Complexity , 1996, CCCG.

[22]  Franz-Josef Brandenburg.  1-Visibility Representations of 1-Planar Graphs , 2014, J. Graph Algorithms Appl..

[23]  Emilio Di Giacomo,et al.  Visibility Representations of Boxes in 2.5 Dimensions , 2016, Graph Drawing.

[24]  Franz-Josef Brandenburg.  Characterizing and Recognizing 4-Map Graphs , 2018, Algorithmica.

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

[26]  Robert E. Tarjan,et al.  Rectilinear planar layouts and bipolar orientations of planar graphs , 1986, Discret. Comput. Geom..

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

[28]  Ellen Gethner,et al.  Bar k-Visibility Graphs , 2007, J. Graph Algorithms Appl..

[29]  Roberto Tamassia,et al.  A unified approach to visibility representations of planar graphs , 1986, Discret. Comput. Geom..