An Experimental Study of a 1-planarity Testing and Embedding Algorithm

The definition of $1$-planar graphs naturally extends graph planarity, namely a graph is $1$-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time, deciding whether a graph is $1$-planar is NP-complete, even for restricted classes of graphs. Although several polynomial-time algorithms have been described for recognizing specific subfamilies of $1$-planar graphs, no implementations of general algorithms are available to date. We investigate the feasibility of a $1$-planarity testing and embedding algorithm based on a backtracking strategy. While the experiments show that our approach can be successfully applied to graphs with up to 30 vertices, they also suggest the need of more sophisticated techniques to attack larger graphs. Our contribution provides initial indications that may stimulate further research on the design of practical approaches for the $1$-planarity testing problem.

[1]  Giuseppe Liotta,et al.  A Linear-Time Algorithm for Testing Outer-1-Planarity , 2013, Algorithmica.

[2]  Stephen G. Kobourov,et al.  On the Book Thickness of 1-Planar Graphs , 2015, ArXiv.

[3]  Franz-Josef Brandenburg,et al.  Recognizing Optimal 1-Planar Graphs in Linear Time , 2016, Algorithmica.

[4]  David Eppstein,et al.  Crossing Patterns in Nonplanar Road Networks , 2017, SIGSPATIAL/GIS.

[5]  Emilio Di Giacomo,et al.  Ortho-Polygon Visibility Representations of Embedded Graphs , 2016, Graph Drawing.

[6]  Michael A. Bekos,et al.  The Book Thickness of 1-Planar Graphs is Constant , 2015, Algorithmica.

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

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

[9]  David Eppstein,et al.  Listing All Maximal Cliques in Large Sparse Real-World Graphs , 2011, JEAL.

[10]  Walter Didimo,et al.  Drawing graphs with right angle crossings , 2009, Theor. Comput. Sci..

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

[12]  M.N.S. Swamy,et al.  O(n2) algorithms for graph planarization , 1989, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[13]  Petr Hliněný,et al.  Inserting Multiple Edges into a Planar Graph , 2015, SoCG.

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

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

[16]  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 .

[17]  Martin Grohe.  Local Tree-Width, Excluded Minors, and Approximation Algorithms , 2003, Comb..

[18]  David Eppstein,et al.  Parameterized Complexity of 1-Planarity , 2013, WADS.

[19]  Giuseppe Liotta,et al.  An Experimental Comparison of Four Graph Drawing Algorithms , 1997, Comput. Geom..

[20]  Petra Mutzel,et al.  Inserting an Edge into a Planar Graph , 2001, SODA '01.

[21]  Ioannis G. Tollis,et al.  Fan-planarity: Properties and complexity , 2015, Theor. Comput. Sci..

[22]  Bojan Mohar,et al.  Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard , 2012, SIAM J. Comput..

[23]  Markus Chimani,et al.  A New Approach to Exact Crossing Minimization , 2008, ESA.

[24]  M. N. Shanmukha Swamy,et al.  O(n²) Algorithms for Graph Planarization , 1988, WG.

[25]  Petra Mutzel,et al.  An Experimental Study of Crossing Minimization Heuristics , 2003, Graph Drawing.

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

[27]  Michael Jünger,et al.  The Open Graph Drawing Framework (OGDF) , 2013, Handbook of Graph Drawing and Visualization.

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

[29]  Weidong Huang,et al.  Larger crossing angles make graphs easier to read , 2014, J. Vis. Lang. Comput..

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

[31]  Hiroshi Nagamochi,et al.  Re-embedding a 1-Plane Graph into a Straight-Line Drawing in Linear Time , 2016, GD.

[32]  Christian Bachmaier,et al.  Outer 1-Planar Graphs , 2016, Algorithmica.

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

[34]  David Eppstein.  Diameter and Treewidth in Minor-Closed Graph Families , 2000, Algorithmica.

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

[36]  János Pach,et al.  Graphs drawn with few crossings per edge , 1996, GD.

[37]  David Eppstein,et al.  Structure of Graphs with Locally Restricted Crossings , 2015, SIAM J. Discret. Math..

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

[39]  Michael A. Bekos,et al.  On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs , 2014, Algorithmica.

[40]  Takeshi Tokuyama,et al.  Algoritihmcs for Beyond Planar Graphs (NII Shonan Meeting 2016-17) , 2016, NII Shonan Meet. Rep..

[41]  Pat Morin,et al.  Planar Graphs have Bounded Queue-Number , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[42]  Michael Kaufmann,et al.  Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452) , 2016, Dagstuhl Reports.