On RAC drawings of 1-planar graphs

Abstract A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k -bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing) . The relationships between 1-planar and k -bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: ( i ) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? ( i i ) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs.

[1]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[2]  Michael Kaufmann,et al.  Journal of Graph Algorithms and Applications on the Perspectives Opened by Right Angle Crossing Drawings , 2022 .

[3]  Giuseppe Liotta,et al.  Graph drawing beyond planarity: some results and open problems , 2014, ICTCS.

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

[5]  Walter Didimo.  Density of straight-line 1-planar graph drawings , 2013, Inf. Process. Lett..

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

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

[8]  Helen C. Purchase,et al.  Effective information visualisation: a study of graph drawing aesthetics and algorithms , 2000, Interact. Comput..

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

[10]  Michael A. Bekos,et al.  The Straight-Line RAC Drawing Problem is NP-Hard , 2010, J. Graph Algorithms Appl..

[11]  Csaba D. Tóth,et al.  Graphs that admit right angle crossing drawings , 2010, Comput. Geom..

[12]  Xin Zhang.  Drawing complete multipartite graphs on the plane with restrictions on crossings , 2013, ArXiv.

[13]  Michael Kaufmann,et al.  The Density of Fan-Planar Graphs , 2014, Electron. J. Comb..

[14]  Weidong Huang,et al.  Using eye tracking to investigate graph layout effects , 2007, 2007 6th International Asia-Pacific Symposium on Visualization.

[15]  TamassiaRoberto,et al.  On-Line Planarity Testing , 1996 .

[16]  Helen C. Purchase,et al.  Which Aesthetic has the Greatest Effect on Human Understanding? , 1997, GD.

[17]  Walter Didimo,et al.  Drawing Graphs with Right Angle Crossings , 2009, WADS.

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

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

[20]  Peter Eades,et al.  Every outer-1-Plane Graph has a Right Angle Crossing Drawing , 2012, Int. J. Comput. Geom. Appl..

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

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

[23]  Roberto Tamassia,et al.  On-Line Planarity Testing , 1989, SIAM J. Comput..

[24]  Peter Eades,et al.  Effects of Crossing Angles , 2008, 2008 IEEE Pacific Visualization Symposium.

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

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

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

[28]  Colin Ware,et al.  Cognitive Measurements of Graph Aesthetics , 2002, Inf. Vis..

[29]  David A. Carrington,et al.  Empirical Evaluation of Aesthetics-based Graph Layout , 2002, Empirical Software Engineering.

[30]  Takao Nishizeki,et al.  Planar Graphs: Theory and Algorithms , 1988 .

[31]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

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

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