Embedding-Preserving Rectangle Visibility Representations of Nonplanar Graphs

A (weak) rectangle visibility representation, or simply an RVR, of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Given a graph with a fixed embedding in the plane, we show that the problem of testing whether this graph has an embedding-preserving RVR can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs, i.e., for embedded graphs having at most one crossing per edge. The linear time algorithm uses three forbidden configurations, which extend the set known for straight-line drawings of 1-plane graphs. The algorithm first checks for the presence of these forbidden configurations in the input graph, and then either an embedding-preserving RVR is computed (also in linear time) or a forbidden configuration is reported as a negative witness. Finally, we discuss extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

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

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

[3]  Éric Fusy,et al.  Transversal structures on triangulations: A combinatorial study and straight-line drawings , 2006, Discret. Math..

[4]  David Eppstein,et al.  On the Density of Maximal 1-Planar Graphs , 2012, Graph Drawing.

[5]  Goos Kant,et al.  A More Compact Visibility Representation , 1997, Int. J. Comput. Geom. Appl..

[6]  Vladimir P. Korzhik.  Minimal non-1-planar graphs , 2008, Discret. Math..

[7]  Stefan Felsner,et al.  Rectangle and Square Representations of Planar Graphs , 2013 .

[8]  Július Czap,et al.  On Drawings and Decompositions of 1-Planar Graphs , 2013, Electron. J. Comb..

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

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

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

[12]  Norishige Chiba,et al.  Arboricity and Subgraph Listing Algorithms , 1985, SIAM J. Comput..

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

[14]  Goos Kant,et al.  Area Requirement of Visibility Representations of Trees , 1997, Inf. Process. Lett..

[15]  Franz-Josef Brandenburg,et al.  Journal of Graph Algorithms and Applications 1-planarity of Graphs with a Rotation System 68 Auer Et Al. 1-planarity of Graphs with a Rotation System , 2022 .

[16]  Walter Didimo,et al.  Recognizing and drawing IC-planar graphs , 2015, Theor. Comput. Sci..

[17]  Xin He,et al.  On Finding the Rectangular Duals of Planar Triangular Graphs , 1993, SIAM J. Comput..

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

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

[20]  Zhi-Zhong Chen,et al.  Recognizing Hole-Free 4-Map Graphs in Cubic Time , 2005, Algorithmica.

[21]  K. Wagner,et al.  Bemerkungen zu einem Sechsfarbenproblem von G. Ringel , 1983 .

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

[23]  Jan Kyncl.  Enumeration of simple complete topological graphs , 2009, Eur. J. Comb..

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

[25]  Goos Kant,et al.  Regular Edge Labeling of 4-Connected Plane Graphs and Its Applications in Graph Drawing Problems , 1997, Theor. Comput. Sci..

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

[27]  Goos Kant,et al.  Triangulating Planar Graphs while Minimizing the Maximum Degree , 1992, Inf. Comput..

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

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

[30]  Piotr Sankowski,et al.  Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in Õ (m10/7 log W) Time (Extended Abstract) , 2016, SODA.

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

[32]  Walter Didimo,et al.  The Crossing-Angle Resolution in Graph Drawing , 2013 .

[33]  Jeffrey D Ullma.  Computational Aspects of VLSI , 1984 .

[34]  Carsten Thomassen,et al.  Rectilinear drawings of graphs , 1988, J. Graph Theory.

[35]  Therese C. Biedl,et al.  Relating Bends and Size in Orthogonal Graph Drawings , 1998, Inf. Process. Lett..

[36]  Anna Lubiw,et al.  Morphing orthogonal planar graph drawings , 2013, SODA '06.

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

[38]  Aleksander Madry,et al.  Computing Maximum Flow with Augmenting Electrical Flows , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[39]  Yahya Ould Hamidoune,et al.  Representing a planar graph by vertical lines joining different levels , 1983, Discret. Math..

[40]  Sue Whitesides,et al.  Rectangle Visibility Graphs: Characterization, Construction, and Compaction , 2003, STACS.

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

[42]  Roberto Tamassia,et al.  On Embedding a Graph in the Grid with the Minimum Number of Bends , 1987, SIAM J. Comput..

[43]  Giuseppe Liotta,et al.  L-Visibility Drawings of IC-Planar Graphs , 2015, Graph Drawing.

[44]  Yusuke Suzuki.  Re-embeddings of Maximum 1-Planar Graphs , 2010, SIAM J. Discret. Math..