The QuaSEFE Problem

We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QuaSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. We show that a triple consisting of two planar graphs and a tree admit a QuaSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QuaSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QuaSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QuaSEFE for two matchings if the quasiplanar drawing of one matching is fixed.

[1]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[2]  Jan Kratochvíl,et al.  Testing planarity of partially embedded graphs , 2010, SODA '10.

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

[4]  Michael A. Bekos.  Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends , 2014 .

[5]  Emilio Di Giacomo,et al.  Planar and Quasi Planar Simultaneous Geometric Embedding , 2014, Graph Drawing.

[6]  János Pach,et al.  The Number of Edges in k-Quasi-planar Graphs , 2011, SIAM J. Discret. Math..

[7]  Okamoto Yoshio,et al.  Vertex Angle and Crossing Angle Resolution of Leveled Tree Drawings , 2011 .

[8]  Micha Sharir,et al.  Quasi-planar graphs have a linear number of edges , 1995, GD.

[9]  Giuseppe Liotta,et al.  Simultaneous Visibility Representations of Plane st-graphs Using L-shapes , 2015, WG.

[10]  F. Brandenburg.  A Simple Quasi-planar Drawing of K10 , 2016 .

[11]  Stephen G. Kobourov,et al.  Simultaneous Embedding of Planar Graphs , 2012, Handbook of Graph Drawing and Visualization.

[12]  János Pach,et al.  Embedding Planar Graphs at Fixed Vertex Locations , 1998, GD.

[13]  Michael Jünger,et al.  Simultaneous Geometric Graph Embeddings , 2007, GD.

[14]  Michael A. Bekos,et al.  Geometric RAC Simultaneous Drawings of Graphs , 2012, J. Graph Algorithms Appl..

[15]  Joseph S. B. Mitchell,et al.  On Simultaneous Planar Graph Embeddings , 2003, WADS.

[16]  Bernhard Haeupler,et al.  Testing Simultaneous Planarity when the Common Graph is 2-Connected , 2010, J. Graph Algorithms Appl..

[17]  Eyal Ackerman.  A note on 1-planar graphs , 2014, Discret. Appl. Math..

[18]  Fabrizio Frati.  Embedding Graphs Simultaneously with Fixed Edges , 2007 .

[19]  Andrzej Pelc,et al.  Gathering Despite Mischief , 2012, SODA.

[20]  Marcus Schaefer,et al.  Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants , 2012, J. Graph Algorithms Appl..

[21]  Michael Kaufmann,et al.  On a Tree and a Path with no Geometric Simultaneous Embedding , 2010, J. Graph Algorithms Appl..

[22]  Gábor Tardos,et al.  On the maximum number of edges in quasi-planar graphs , 2007, J. Comb. Theory, Ser. A.

[23]  Ignaz Rutter,et al.  Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems , 2013, SODA.

[24]  Giordano Da Lozzo,et al.  Advancements on SEFE and Partitioned Book Embedding problems , 2013, Theor. Comput. Sci..

[25]  Luca Grilli.  On the NP-hardness of GRacSim drawing and k-SEFE Problems , 2018, J. Graph Algorithms Appl..