survey

A Survey on Graph Drawing Beyond Planarity

Authors:

Giuseppe Liotta,

Fabrizio MontecchianiAuthors Info & Claims

ACM Computing Surveys (CSUR), Volume 52, Issue 1

Article No.: 4, Pages 1 - 37

https://doi.org/10.1145/3301281

Published: 21 February 2019 Publication History

Abstract

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of nonplanar graphs in terms of forbidden crossing configurations. The aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

References

[1]

Eyal Ackerman. 2009. On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discr. Comput. Geom. 41, 3 (2009), 365--375.

Digital Library

[2]

Eyal Ackerman. 2014. A note on 1-planar graphs. Discr. Appl. Math. 175 (2014), 104--108.

[3]

Eyal Ackerman. 2015. On topological graphs with at most four crossings per edge. CoRR abs/1509.01932 (2015).

[4]

Eyal Ackerman, Jacob Fox, János Pach, and Andrew Suk. 2014. On grids in topological graphs. Comput. Geom. 47, 7 (2014), 710--723.

[5]

Eyal Ackerman, Radoslav Fulek, and Csaba D. Tóth. 2010. On the size of graphs that admit polyline drawings with few bends and crossing angles. In Proceedings of the GD 2010 (LNCS), Vol. 6502. Springer, 1--12. https://link.springer.com/chapter/10.1007/978-3-642-18469-7_1.

Digital Library

[6]

Eyal Ackerman, Radoslav Fulek, and Csaba D. Tóth. 2012. Graphs that admit polyline drawings with few crossing angles. SIAM J. Discr. Math. 26, 1 (2012), 305--320.

Digital Library

[7]

Eyal Ackerman, Balązs Keszegh, and Mate Vizer. 2018. On the size of planarly connected crossing graphs. J. Graph Algorithms Appl. 22, 1 (2018), 11--22.

[8]

Eyal Ackerman and Gábor Tardos. 2007. On the maximum number of edges in quasi-planar graphs. J. Comb. Theory, Ser. A 114, 3 (2007), 563--571.

Digital Library

[9]

Jin Akiyama and Noga Alon. 1989. Disjoint simplices and geometric hypergraphs. Ann. New York Acad. Sci. 555, 1 (1989), 1--3.

[10]

Md. Jawaherul Alam, Franz J. Brandenburg, and Stephen G. Kobourov. 2013. Straight-line grid drawings of 3-connected 1-planar graphs. In Proceedings of the GD 2013 (LNCS), Vol. 8242. Springer, 83--94. https://link.springer.com/chapter/10.1007/978-3-319-03841-4_8.

Digital Library

[11]

Md. Jawaherul Alam, Franz J. Brandenburg, and Stephen G. Kobourov. 2015. On the book thickness of 1-planar graphs. CoRR abs/1510.05891 (2015).

[12]

Md. Jawaherul Alam, William S. Evans, Stephen G. Kobourov, Sergey Pupyrev, Jackson Toeniskoetter, and Torsten Ueckerdt. 2015. Contact representations of graphs in 3D. In Proceedings of the WADS 2015 (LNCS), Vol. 9214. Springer, 14--27. https://link.springer.com/chapter/10.1007%2F978-3-319-21840-3_2.

[13]

Noga Alon and Paul Erdös. 1989. Disjoint edges in geometric graphs. Discr. Comput. Geom. 4 (1989), 287--290.

Digital Library

[14]

Patrizio Angelini, Michael A. Bekos, Franz J. Brandenburg, Giordano Da Lozzo, Giuseppe Di Battista, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Ignaz Rutter. 2017. On the relationship between k-planar and k-quasi-planar graphs. In Proceedings of theWG 2017 (LNCS), Vol. 10520. Springer, 59--74. https://link.springer.com/chapter/10.1007%2F978-3-319-68705-6_5.

[15]

Patrizio Angelini, Michael A. Bekos, Henry Förster, and Michael Kaufmann. 2018. On RAC drawings of graphs with one bend per edge. CoRR abs/1808.10470 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_9.

[16]

Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Philipp Kindermann, and Thomas Schneck. 2018. 1-fan-bundle-planar drawings of graphs. Theor. Comput. Sci. 723 (2018), 23--50.

[17]

Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, and Fabrizio Montecchiani. 2017. 3D visibility representations of 1-planar graphs. In Proceedings of the GD 2017 (LNCS), Vol. 10692. Springer, 102--109. https://link.springer.com/chapter/10.1007%2F978-3-319-73915-1_9.

[18]

Patrizio Angelini, Luca Cittadini, Walter Didimo, Fabrizio Frati, Giuseppe Di Battista, Michael Kaufmann, and Antonios Symvonis. 2011. On the perspectives opened by right angle crossing drawings. J. Graph Algorithms Appl. 15, 1 (2011), 53--78.

[19]

Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. 2015. Relaxing the constraints of clustered planarity. Comput. Geom. 48, 2 (2015), 42--75.

Digital Library

[20]

Patrizio Angelini, Giuseppe Di Battista, Walter Didimo, Fabrizio Frati, Seok-Hee Hong, Michael Kaufmann, Giuseppe Liotta, and Anna Lubiw. 2012. Large angle crossing drawings of planar graphs in subquadratic area. In Proceedings of the EGC 2011 (LNCS), Vol. 7579. Springer, 200--209. https://link.springer.com/chapter/10.1007%2F978-3-642-34191-5_19.

[21]

Patrizio Angelini, Peter Eades, Seok-Hee Hong, Karsten Klein, Stephen G. Kobourov, Giuseppe Liotta, Alfredo Navarra, and Alessandra Tappini. 2018. Turning cliques into paths to achieve planarity. CoRR abs/1808.08925 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_5.

[22]

Patrizio Angelini, Markus Geyer, Michael Kaufmann, and Daniel Neuwirth. 2012. On a tree and a path with no geometric simultaneous embedding. J. Graph Algorithms Appl. 16, 1 (2012), 37--83.

[23]

Arthur Appel, F. James Rohlf, and Arthur J. Stein. 1979. The haloed line effect for hidden line elimination. In Proceedings of the SIGGRAPH 1979. ACM, 151--157. https://dl.acm.org/citation.cfm?doid=800249.807437.

Digital Library

[24]

Evmorfia N. Argyriou, Michael A. Bekos, Michael Kaufmann, and Antonios Symvonis. 2013. Geometric RAC simultaneous drawings of graphs. J. Graph Algorithms Appl. 17, 1 (2013), 11--34.

[25]

Evmorfia N. Argyriou, Michael A. Bekos, and Antonios Symvonis. 2012. The straight-line RAC drawing problem is NP-hard. J. Graph Algorithms Appl. 16, 2 (2012), 569--597.

[26]

Evmorfia N. Argyriou, Michael A. Bekos, and Antonios Symvonis. 2013. Maximizing the total resolution of graphs. Comput. J. 56, 7 (2013), 887--900.

[27]

Evmorfia N. Argyriou, Sabine Cornelsen, Henry Förster, Michael Kaufmann, Martin Nöllenburg, Yoshio Okamoto, Chrysanthi N. Raftopoulou, and Alexander Wolff. 2018. Orthogonal and smooth orthogonal layouts of 1-planar graphs with low edge complexity. CoRR abs/1808.10536 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_36.

[28]

Karin Arikushi, Radoslav Fulek, Balázs Keszegh, Filip Moric, and Csaba D. Tóth. 2012. Graphs that admit right angle crossing drawings. Comput. Geom. 45, 4 (2012), 169--177.

Digital Library

[29]

Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. 2016. Outer 1-planar graphs. Algorithmica 74, 4 (2016), 1293--1320.

Digital Library

[30]

Christopher Auer, Franz-Josef Brandenburg, Andreas Gleißner, and Kathrin Hanauer. 2012. On sparse maximal 2-planar graphs. In Graph Drawing (LNCS), Vol. 7704. Springer, 555--556. https://link.springer.com/chapter/10.1007%2F978-3-642-36763-2_50.

Digital Library

[31]

Christopher Auer, Franz J. Brandenburg, Andreas Gleißner, and Josef Reislhuber. 2015. 1-planarity of graphs with a rotation system. J. Graph Algorithms Appl. 19, 1 (2015), 67--86.

[32]

S. Avital and H. Hanani. 1966. Graphs. Gilyonot Lematematika 3 (1966), 2--8.

[33]

Christian Bachmaier, Franz J. Brandenburg, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. 2017. NIC-planar graphs. Discr. Appl. Math. 232 (2017), 23--40.

Digital Library

[34]

Sang Won Bae, Jean-François Baffier, Jinhee Chun, Peter Eades, Kord Eickmeyer, Luca Grilli, Seok-Hee Hong, Matias Korman, Fabrizio Montecchiani, Ignaz Rutter, and Csaba D. Tóth. 2018. Gap-planar graphs. Theor. Comput. Sci. 745 (2018), 36--52.

[35]

Michael J. Bannister, Sergio Cabello, and David Eppstein. 2018. Parameterized complexity of 1-planarity. J. Graph Algorithms Appl. 22, 1 (2018), 23--49.

[36]

Imre Bárány and Günter Rote. 2005. Strictly convex drawings of planar graphs. CoRR abs/cs/0507030 (2005).

[37]

Vladimir Batagelj, Franz-Josef Brandenburg, Walter Didimo, Giuseppe Liotta, Pietro Palladino, and Maurizio Patrignani. 2011. Visual analysis of large graphs using (X, Y)-clustering and hybrid visualizations. IEEE Trans. Vis. Comput. Graph. 17, 11 (2011), 1587--1598.

Digital Library

[38]

Carlo Batini, L. Furlani, and Enrico Nardelli. 1985. What is a good diagram? A pragmatic approach. In Proceedings of the Fourth International Conference on Entity-Relationship Approach: The Use of {ER} Concept in Knowledge Representation. IEEE Computer Society and North-Holland, 312--319.

Digital Library

[39]

Carlo Batini, Enrico Nardelli, and Roberto Tamassia. 1986. A layout algorithm for data flow diagrams. IEEE Trans. Software Eng. 12, 4 (1986), 538--546.

Digital Library

[40]

Richard A. Becker, Stephen G. Eick, and Allan R. Wilks. 1995. Visualizing network data. IEEE Trans. Vis. Comput. Graph. 1, 1 (1995), 16--28.

Digital Library

[41]

Michael A. Bekos, Till Bruckdorfer, Michael Kaufmann, and Chrysanthi N. Raftopoulou. 2017. The book thickness of 1-planar graphs is constant. Algorithmica 79, 2 (2017), 444--465.

Digital Library

[42]

Michael A. Bekos, Sabine Cornelsen, Luca Grilli, Seok-Hee Hong, and Michael Kaufmann. 2017. On the recognition of fan-planar and maximal outer-fan-planar graphs. Algorithmica 79, 2 (2017), 401--427.

Digital Library

[43]

Michael A. Bekos, Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Chrysanthi N. Raftopoulou. 2018. Edge partitions of optimal 2-plane and 3-plane graphs. In Proceedings of the WG 2018 (LNCS), Vol. 11159. Springer, 27--39. https://link.springer.com/chapter/10.1007%2F978-3-030-00256-5_3.

[44]

Michael A. Bekos, Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, and Fabrizio Montecchiani. 2017. On RAC drawings of 1-planar graphs. Theor. Comput. Sci. 689 (2017), 48--57.

[45]

Michael A. Bekos, Henry Förster, Christian Geckeler, Lukas Holländer, Michael Kaufmann, Amadäus M. Spallek, and Jan Splett. 2018. A heuristic approach towards drawings of graphs with high crossing resolution. CoRR abs/1808.10519 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_19.

[46]

Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. 2016. On the density of non-simple 3-planar graphs. In Proceedings of the GD 2016 (LNCS), Vol. 9801. Springer, 344--356. https://link.springer.com/chapter/10.1007%2F978-3-319-50106-2_27.

[47]

Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. 2017. On optimal 2- and 3-planar graphs. In Proceedings of the SoCG 2017 (LIPIcs), Vol. 77. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 16:1--16:16. http://drops.dagstuhl.de/opus/volltexte/2017/7230/.

[48]

Michael A. Bekos, Thomas C. van Dijk, Philipp Kindermann, and Alexander Wolff. 2016. Simultaneous drawing of planar graphs with right-angle crossings and few bends. J. Graph Algorithms Appl. 20, 1 (2016), 133--158.

[49]

F. Bernhart and P. C. Kainen. 1979. The book thickness of a graph. J. Combin. Theory Ser. B 27 (1979), 320--331.

[50]

Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. 2002. Quasi-upward planarity. Algorithmica 32, 3 (2002), 474--506.

Digital Library

[51]

Therese C. Biedl, Giuseppe Liotta, and Fabrizio Montecchiani. 2018. Embedding-preserving rectangle visibility representations of nonplanar graphs. Discrete and Computational Geometry 60, 2 (2018), 345--380.

Digital Library

[52]

Carla Binucci, Markus Chimani, Walter Didimo, Martin Gronemann, Karsten Klein, Jan Kratochvíl, Fabrizio Montecchiani, and Ioannis G. Tollis. 2017. Algorithms and characterizations for 2-layer fan-planarity: From caterpillar to stegosaurus. J. Graph Algorithms Appl. 21, 1 (2017), 81--102.

[53]

Carla Binucci, Emilio Di Giacomo, Walter Didimo, Fabrizio Montecchiani, Maurizio Patrignani, Antonios Symvonis, and Ioannis G. Tollis. 2015. Fan-planarity: Properties and complexity. Theor. Comput. Sci. 589 (2015), 76--86.

Digital Library

[54]

Carla Binucci, Emilio Di Giacomo, Md. Iqbal Hossain, and Giuseppe Liotta. 2018. 1-page and 2-page drawings with bounded number of crossings per edge. Eur. J. Comb. 68 (2018), 24--37.

[55]

Carla Binucci, Walter Didimo, and Maurizio Patrignani. 2014. Upward and quasi-upward planarity testing of embedded mixed graphs. Theor. Comput. Sci. 526 (2014), 75--89.

Digital Library

[56]

Carla Binucci, Giuseppe Liotta, Fabrizio Montecchiani, and Alessandra Tappini. 2016. Partial edge drawing: Homogeneity is more important than crossings and ink. In Proceedings of the IISA. IEEE, 1--6. https://ieeexplore.ieee.org/document/7785427.

[57]

Thomas Bläsius, Stephen G. Kobourov, and Ignaz Rutter. 2013. Simultaneous embedding of planar graphs. In Handbook on Graph Drawing and Visualization., Roberto Tamassia (Ed.). Chapman and Hall/CRC, 349--381.

[58]

R Bodendiek, H Schumacher, and K Wagner. 1983. Bemerkungen zu einem Sechsfarbenproblem von G. Ringel. Abhandlungen Aus Dem Mathematischen Seminar Der Universitaet Hamburg 53, 1 (1983), 41--52.

[59]

Béla Bollobás. 1978. Extremal Graph Theory. Academic Press, New York.

[60]

Romain Bourqui, David Auber, and Patrick Mary. 2007. How to draw clustered weighted graphs using a multilevel force-directed graph drawing algorithm. In Proceedings of the 11th International Conference on Information Visualisation (IV'07). IEEE, 757--764.

Digital Library

[61]

Franz Brandenburg. 2018. Recognizing IC-planar and NIC-planar graphs. J. Graph Algorithms Appl. 22, 2 (2018), 239--271.

[62]

Franz-Josef Brandenburg, David Eppstein, Andreas Gleißner, Michael T. Goodrich, Kathrin Hanauer, and Josef Reislhuber. 2012. On the density of maximal 1-planar graphs. In Proceedings of the GD 2012 (LNCS), Vol. 7704. Springer, 327--338. https://link.springer.com/chapter/10.1007%2F978-3-642-36763-2_29.

Digital Library

[63]

Franz J. Brandenburg. 2014. 1-visibility representations of 1-planar graphs. J. Graph Algorithms Appl. 18, 3 (2014), 421--438.

[64]

Franz J. Brandenburg. 2018. A first order logic definition of beyond-planar graphs. J. Graph Algorithms Appl. 22, 1 (2018), 51--66.

[65]

Franz J. Brandenburg. 2018. On fan-crossing and fan-crossing free graphs. Inf. Process. Lett. 138 (2018), 67--71.

[66]

Franz J. Brandenburg. 2018. Recognizing optimal 1-planar graphs in linear time. Algorithmica 80, 1 (2018), 1--28.

Digital Library

[67]

Franz J. Brandenburg, Walter Didimo, William S. Evans, Philipp Kindermann, Giuseppe Liotta, and Fabrizio Montecchiani. 2016. Recognizing and drawing IC-planar graphs. Theor. Comput. Sci. 636 (2016), 1--16.

Digital Library

[68]

Peter Braß, Eowyn Cenek, Christian A. Duncan, Alon Efrat, Cesim Erten, Dan Ismailescu, Stephen G. Kobourov, Anna Lubiw, and Joseph S. B. Mitchell. 2007. On simultaneous planar graph embeddings. Comput. Geom. 36, 2 (2007), 117--130.

Digital Library

[69]

Peter Brass, Gyula Károlyi, and Pavel Valtr. 2003. A Turán-type extremal theorey of convex geometric graphs. Discr. Comput. Geom. 25 (2003), 275--300.

[70]

Peter Braß, William O. J. Moser, and János Pach. 2005. Research Problems in Discrete Geometry. Springer.

[71]

Till Bruckdorfer, Sabine Cornelsen, Carsten Gutwenger, Michael Kaufmann, Fabrizio Montecchiani, Martin Nöllenburg, and Alexander Wolff. 2017. Progress on partial edge drawings. J. Graph Algorithms Appl. 21, 4 (2017), 757--786.

[72]

Till Bruckdorfer and Michael Kaufmann. 2012. Mad at edge crossings? Break the edges&excl;. In Proceedings of the FUN 2012 (LNCS), Vol. 7288. Springer, 40--50. https://link.springer.com/chapter/10.1007%2F978-3-642-30347-0_7.

Digital Library

[73]

Till Bruckdorfer, Michael Kaufmann, and Andreas Lauer. 2015. A practical approach for 1/4-SHPEDs. In Proceedings of the IISA 2015. IEEE, 1--6. https://ieeexplore.ieee.org/document/7387994?arnumber=7387994.

[74]

Till Bruckdorfer, Michael Kaufmann, and Fabrizio Montecchiani. 2014. 1-bend orthogonal partial edge drawing. J. Graph Algorithms Appl. 18, 1 (2014), 111--131.

[75]

Christoph Buchheim, Markus Chimani, Carsten Gutwenger, Michael Jünger, and Petra Mutzel. 2013. Crossings and planarization. In Handbook of Graph Drawing and Visualization, Roberto Tamassia (Ed.). Chapman and Hall/CRC, 43--85.

[76]

Sergio Cabello and Bojan Mohar. 2011. Crossing number and weighted crossing number of near-planar graphs. Algorithmica 60, 3 (2011), 484--504.

Digital Library

[77]

Sergio Cabello and Bojan Mohar. 2013. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42, 5 (2013), 1803--1829.

[78]

Vasilis Capoyleas and János Pach. 1992. A turán-type theorem on chords of a convex polygon. J. Comb. Theory, Ser. B 56, 1 (1992), 9--15.

Digital Library

[79]

Marie-Jose Carpano. 1980. Automatic display of hierarchized graphs for computer-aided decision analysis. IEEE Trans. Systems, Man, and Cybernetics 10, 11 (1980), 705--715.

[80]

Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, and Alexander Wolff. 2017. Beyond outerplanarity. In Proceedings of the GD 2017 (LNCS), Vol. 10692. Springer, 546--559. https://link.springer.com/chapter/10.1007%2F978-3-319-73915-1_42.

[81]

Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink. 2018. Compact drawings of 1-planar graphs with right-angle crossings and few bends. CoRR abs/1806.10044 (2018). Accepted at GD 2018.

[82]

Otfried Cheong, Sariel Har-Peled, Heuna Kim, and Hyo-Sil Kim. 2015. On the number of edges of fan-crossing free graphs. Algorithmica 73, 4 (2015), 673--695.

Digital Library

[83]

Markus Chimani, Carsten Gutwenger, Petra Mutzel, and Christian Wolf. 2009. Inserting a vertex into a planar graph. In Proceedings of the ACM-SIAM SoDA 2009. SIAM, 375--383. https://dl.acm.org/citation.cfm?id=1496770.1496812.

Digital Library

[84]

Markus Chimani and Petr Hlinený. 2016. Inserting multiple edges into a planar graph. In Proceedings of the SoCG 2016 (LIPIcs), Vol. 51. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 30:1--30:15. http://drops.dagstuhl.de/opus/volltexte/2016/5922/.

[85]

Pier Francesco Cortese and Giuseppe Di Battista. 2005. Clustered planarity. In Proceedings of the ACM SoCG 2005. ACM, 32--34. https://dl.acm.org/citation.cfm?doid=1064092.1064093.

Digital Library

[86]

J. Czap and P. Šugerek. 2017. Drawing graph joins in the plane with restrictions on crossings. Filomat 31, 2 (2017), 363--370.

[87]

Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignani. 2018. Computing NodeTrix representations of clustered graphs. J. Graph Algorithms Appl. 22, 2 (2018), 139--176.

[88]

Hubert de Fraysseix, János Pach, and Richard Pollack. 1990. How to draw a planar graph on a grid. Combinatorica 10, 1 (1990), 41--51.

[89]

Alice M. Dean, William S. Evans, Ellen Gethner, Joshua D. Laison, Mohammad Ali Safari, and William T. Trotter. 2007. Bar -visibility graphs. J. Graph Algorithms Appl. 11, 1 (2007), 45--59.

[90]

Hooman Reisi Dehkordi and Peter Eades. 2012. Every outer-1-plane graph has a right angle crossing drawing. Int. J. Comput. Geometry Appl. 22, 6 (2012), 543--558.

[91]

Hooman Reisi Dehkordi, Peter Eades, Seok-Hee Hong, and Quan Hoang Nguyen. 2016. Circular right-angle crossing drawings in linear time. Theor. Comput. Sci. 639 (2016), 26--41.

Digital Library

[92]

Erik D. Demaine and Mohammad Taghi Hajiaghayi. 2004. Diameter and treewidth in minor-closed graph families, revisited. Algorithmica 40, 3 (2004), 211--215.

[93]

Almut Demel, Dominik Dürrschnabel, Tamara Mchedlidze, Marcel Radermacher, and Lasse Wulf. 2018. A greedy heuristic for crossing-angle maximization. CoRR abs/1807.09483 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_20.

[94]

Giuseppe Di Battista, Walter Didimo, and A. Marcandalli. 2002. Planarization of clustered graphs. In Proceedings of the GD 2001 (LNCS), Vol. 2265. Springer, 60--74. https://link.springer.com/chapter/10.1007%2F3-540-45848-4_5.

[95]

Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. 1999. Graph Drawing. Prentice Hall, Upper Saddle River, NJ.

Digital Library

[96]

Giuseppe Di Battista and Roberto Tamassia. 1988. Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61 (1988), 175--198.

Digital Library

[97]

Giuseppe Di Battista, Roberto Tamassia, and Ioannis G. Tollis. 1992. Area requirement and symmetry display of planar upward drawings. Discr. Comput. Geom. 7 (1992), 381--401.

Digital Library

[98]

Emilio Di Giacomo, Walter Didimo, Peter Eades, and Giuseppe Liotta. 2014. 2-layer right angle crossing drawings. Algorithmica 68, 4 (2014), 954--997.

[99]

Emilio Di Giacomo, Walter Didimo, William S. Evans, Giuseppe Liotta, Henk Meijer, Fabrizio Montecchiani, and Stephen K. Wismath. 2018. Ortho-polygon visibility representations of embedded graphs. Algorithmica 80, 8 (2018), 2345--2383.

Digital Library

[100]

Emilio Di Giacomo, Walter Didimo, William S. Evans, Giuseppe Liotta, Henk Meijer, Fabrizio Montecchiani, and Stephen K. Wismath. 2018. New results on edge partitions of 1-plane graphs. Theor. Comput. Sci. 713 (2018), 78--84.

[101]

Emilio Di Giacomo, Walter Didimo, Luca Grilli, Giuseppe Liotta, and Salvatore Agostino Romeo. 2015. Heuristics for the maximum 2-layer RAC subgraph problem. Comput. J. 58, 5 (2015), 1085--1098.

[102]

Emilio Di Giacomo, Walter Didimo, and Giuseppe Liotta. 2013. Spine and radial drawings. In Handbook on Graph Drawing and Visualization., Roberto Tamassia (Ed.). Chapman and Hall/CRC, 247--284.

[103]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Henk Meijer. 2011. Area, curve complexity, and crossing resolution of non-planar graph drawings. Theory Comput. Syst. 49, 3 (2011), 565--575.

Digital Library

[104]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, and Stephen K. Wismath. 2015. Planar and quasi-planar simultaneous geometric embedding. Comput. J. 58, 11 (2015), 3126--3140.

[105]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. 2012. h-quasi planar drawings of bounded treewidth graphs in linear area. In Proceedings of the WG 2012 (LNCS), Vol. 7551. Springer, 91--102. https://link.springer.com/chapter/10.1007%2F978-3-642-34611-8_12.

Digital Library

[106]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. 2013. Area requirement of graph drawings with few crossings per edge. Comput. Geom. 46, 8 (2013), 909--916.

[107]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. 2017. Area-thickness trade-offs for straight-line drawings of planar graphs. Comput. J. 60, 1 (2017), 135--142.

[108]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Ioannis G. Tollis. 2014. Techniques for edge stratification of complex graph drawings. J. Vis. Lang. Comput. 25, 4 (2014), 533--543.

Digital Library

[109]

Emilio Di Giacomo and Giuseppe Liotta. 2007. Simultaneous embedding of outerplanar graphs, paths, and cycles. Int. J. Comput. Geometry Appl. 17, 2 (2007), 139--160.

[110]

Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. 2015. Drawing outer 1-planar graphs with few slopes. J. Graph Algorithms Appl. 19, 2 (2015), 707--741.

[111]

Emilio Di Giacomo, Giuseppe Liotta, Maurizio Patrignani, and Alessandra Tappini. 2018. NodeTrix planarity testing with small clusters. In Proceedings of the GD 2017 (LNCS), Vol. 10692. Springer, 479--491. https://link.springer.com/chapter/10.1007%2F978-3-319-73915-1_37.

[112]

Emilio Di Giacomo, Giuseppe Liotta, Maurizio Patrignani, and Alessandra Tappini. 2018. Planar k-NodeTrix graphs a new family of beyond planar graphs. In Proceedings of the GD 2017 (LNCS), Vol. 10692. Springer, 609--611. https://link.springer.com/book/10.1007/978-3-319-73915-1?page=2#toc.

[113]

Josep Díaz, Jordi Petit, and Maria J. Serna. 2002. A survey of graph layout problems. ACM Comput. Surv. 34, 3 (2002), 313--356.

Digital Library

[114]

Matthew Dickerson, David Eppstein, Michael T. Goodrich, and Jeremy Yu Meng. 2005. Confluent drawings: Visualizing non-planar diagrams in a planar way. J. Graph Algorithms Appl. 9, 1 (2005), 31--52.

[115]

Walter Didimo. 2013. Density of straight-line 1-planar graph drawings. Inf. Process. Lett. 113, 7 (2013), 236--240.

Digital Library

[116]

Walter Didimo. 2016. Upward graph drawing. In Encyclopedia of Algorithms. 2308--2312.

[117]

Walter Didimo, Peter Eades, and Giuseppe Liotta. 2010. A characterization of complete bipartite RAC graphs. Inf. Process. Lett. 110, 16 (2010), 687--691.

Digital Library

[118]

Walter Didimo, Peter Eades, and Giuseppe Liotta. 2011. Drawing graphs with right angle crossings. Theor. Comput. Sci. 412, 39 (2011), 5156--5166.

[119]

Walter Didimo and Giuseppe Liotta. 2013. The crossing-angle resolution in graph drawing. In Thirty Essays on Geometric Graph Theory, Janos Pach (Ed.). Springer, New York, NY, 167--184.

[120]

Walter Didimo, Giuseppe Liotta, and Salvatore Agostino Romeo. 2011. A graph drawing application to web site traffic analysis. J. Graph Algorithms Appl. 15, 2 (2011), 229--251.

[121]

Walter Didimo, Giuseppe Liotta, and Salvatore Agostino Romeo. 2011. Topology-driven force-directed algorithms. In Proceedings of the GD 2010 (LNCS), Vol. 6502. Springer, 165--176. https://link.springer.com/chapter/10.1007%2F978-3-642-18469-7_15.

Digital Library

[122]

Walter Didimo and Fabrizio Montecchiani. 2014. Fast layout computation of clustered networks: Algorithmic advances and experimental analysis. Inf. Sci. 260 (2014), 185--199.

Digital Library

[123]

U. Dogrusöz, E. Giral, A. Cetintas, A. Civril, and E. Demir. 2009. A layout algorithm for undirected compound graphs. Inf. Sci. 179, 7 (2009), 980--994.

Digital Library

[124]

Andreas W. M. Dress, Jack H. Koolen, and Vincent Moulton. 2002. On line arrangements in the hyperbolic plane. Eur. J. Comb. 23, 5 (2002), 549--557.

Digital Library

[125]

Vida Dujmovic. 2015. Graph layouts via layered separators. J. Comb. Theory, Ser. B 110 (2015), 79--89.

Digital Library

[126]

Vida Dujmovic, David Eppstein, and David R. Wood. 2017. Structure of graphs with locally restricted crossings. SIAM J. Discr. Math. 31, 2 (2017), 805--824.

[127]

Vida Dujmovic and Fabrizio Frati. 2018. Stack and queue layouts via layered separators. J. Graph Algorithms Appl. 22, 1 (2018), 89--99.

[128]

Vida Dujmovic, Joachim Gudmundsson, Pat Morin, and Thomas Wolle. 2011. Notes on large angle crossing graphs. Chicago J. Theor. Comput. Sci. 2011 (2011).

[129]

Vida Dujmovic, Pat Morin, and David R. Wood. 2017. Layered separators in minor-closed graph classes with applications. J. Comb. Theory, Ser. B 127 (2017), 111--147.

[130]

Vida Dujmovic and David R. Wood. 2004. On linear layouts of graphs. Discr. Math. 8 Theor. Comp. Sci. 6, 2 (2004), 339--358.

[131]

Vida Dujmovic and David R. Wood. 2005. Stacks, queues and tracks: Layouts of graph subdivisions. Discr. Math. 8 Theor. Comp. Sci. 7, 1 (2005), 155--202.

[132]

Peter Eades, Seok-Hee Hong, Giuseppe Liotta, Naoki Katoh, and Sheung-Hung Poon. 2015. Straight-line drawability of a planar graph plus an edge. In Proceedings of the WADS 2015 (LNCS), Vol. 9214. Springer, 301--313. https://link.springer.com/chapter/10.1007%2F978-3-319-21840-3_25.

[133]

Peter Eades and Giuseppe Liotta. 2013. Right angle crossing graphs and 1-planarity. Discr. Appl. Math. 161, 7--8 (2013), 961--969.

Digital Library

[134]

P. Eades, B. McKay, and N. Wormald. 1986. On an edge crossing problem. In Proceedings of the ACSC 1986. 327--334. http://users.cecs.anu.edu.au/&sim;bdm/papers/EdgeCrossing.pdf.

[135]

Peter Eades and Nicholas C. Wormald. 1994. Edge crossings in drawings of bipartite graphs. Algorithmica 11, 4 (1994), 379--403.

Digital Library

[136]

David Eppstein. 2000. Diameter and treewidth in minor-closed graph families. Algorithmica 27, 3 (2000), 275--291.

[137]

David Eppstein, Michael T. Goodrich, and Jeremy Yu Meng. 2007. Confluent layered drawings. Algorithmica 47, 4 (2007), 439--452.

[138]

David Eppstein, Marc J. van Kreveld, Elena Mumford, and Bettina Speckmann. 2009. Edges and switches, tunnels and bridges. Comput. Geom. 42, 8 (2009), 790--802.

Digital Library

[139]

William S. Evans, Michael Kaufmann, William Lenhart, Tamara Mchedlidze, and Stephen K. Wismath. 2014. Bar 1-visibility graphs vs. other nearly planar graphs. J. Graph Algorithms Appl. 18, 5 (2014), 721--739.

[140]

William S. Evans, Giuseppe Liotta, and Fabrizio Montecchiani. 2016. Simultaneous visibility representations of plane st-graphs using L-shapes. Theor. Comput. Sci. 645 (2016), 100--111.

Digital Library

[141]

István Fáry. 1948. On straight-line representation of planar graphs. Acta Sci. Math. 11 (1948), 229--233.

[142]

Martin Fink, Jan-Henrik Haunert, Tamara Mchedlidze, Joachim Spoerhase, and Alexander Wolff. 2012. Drawing graphs with vertices at specified positions and crossings at large angles. In Proceedings of the WALCOM (LNCS), Vol. 7157. Springer, 186--197. https://link.springer.com/chapter/10.1007%2F978-3-642-28076-4_19.

Digital Library

[143]

Jacob Fox, János Pach, and Andrew Suk. 2013. The number of edges in k-quasi-planar graphs. SIAM J. Discr. Math. 27, 1 (2013), 550--561.

[144]

M. R. Garey and D. S. Johnson. 1983. Crossing number is NP-complete. SIAM J. Alg. Discr. Meth. 4 (1983), 312--316.

Digital Library

[145]

Ashim Garg and Roberto Tamassia. 2001. On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31, 2 (2001), 601--625.

Digital Library

[146]

Jesse Geneson, Tanya Khovanova, and Jonathan Tidor. 2014. Convex geometric (k+2)-quasiplanar representations of semi-bar k-visibility graphs. Discr. Math. 331 (2014), 83--88.

[147]

Alexander Grigoriev and Hans L. Bodlaender. 2007. Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49, 1 (2007), 1--11.

Digital Library

[148]

Luca Grilli. 2018. On the NP-hardness of GRacSim drawing and k-SEFE Problems. J. Graph Algorithms Appl. 22, 1 (2018), 101--116.

[149]

Arvind Gupta and Russell Impagliazzo. 1991. Computing planar intertwines. In Proceedings of the IEEE FoCS 1991. IEEE, 802--811. https://ieeexplore.ieee.org/document/185452.

Digital Library

[150]

Carsten Gutwenger, Petra Mutzel, and René Weiskircher. 2005. Inserting an edge into a planar graph. Algorithmica 41, 4 (2005), 289--308.

[151]

Nathalie Henry, Jean-Daniel Fekete, and Michael J. McGuffin. 2007. NodeTrix: A hybrid visualization of social networks. IEEE Trans. Vis. Comput. Graph. 13, 6 (2007), 1302--1309.

Digital Library

[152]

Michael Hoffmann and Csaba D. Tóth. 2017. Two-planar graphs are quasiplanar. In Proceedings of the MFCS 2017 (LIPIcs), Vol. 83. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 47:1--47:14. http://drops.dagstuhl.de/opus/volltexte/2017/8081/.

[153]

Danny Holten. 2006. Hierarchical edge bundles: Visualization of adjacency relations in hierarchical data. IEEE Trans. Vis. Comput. Graph. 12, 5 (2006), 741--748.

Digital Library

[154]

Danny Holten and Jarke J. van Wijk. 2009. Force-directed edge bundling for graph visualization. Comput. Graph. Forum 28, 3 (2009), 983--990.

Digital Library

[155]

Seok-Hee Hong, Peter Eades, Naoki Katoh, Giuseppe Liotta, Pascal Schweitzer, and Yusuke Suzuki. 2015. A linear-time algorithm for testing outer-1-planarity. Algorithmica 72, 4 (2015), 1033--1054.

Digital Library

[156]

Seok-Hee Hong and Hiroshi Nagamochi. 2016. Re-embedding a 1-plane graph into a straight-line drawing in linear time. In Proceedings of the GD 2016 (LNCS), Vol. 9801. Springer, 321--334. https://link.springer.com/chapter/10.1007%2F978-3-319-50106-2_25.

[157]

Seok-Hee Hong and Hiroshi Nagamochi. 2016. Testing full outer-2-planarity in linear time. In Proceedings of the WG 2015 (LNCS), Vol. 9224. Springer, 406--421. https://link.springer.com/chapter/10.1007%2F978-3-662-53174-7_29.

Digital Library

[158]

Seok-Hee Hong, Peter Eades, Giuseppe Liotta, and Sheung-Hung Poon. 2012. Fáry’s theorem for 1-planar graphs. In COCOON 2012 (LNCS), Vol. 7434. Springer, 335--346. https://link.springer.com/chapter/10.1007%2F978-3-642-32241-9_29.

[159]

John E. Hopcroft and Robert Endre Tarjan. 1974. Efficient planarity testing. J. ACM 21, 4 (1974), 549--568.

Digital Library

[160]

Weidong Huang. 2007. Using eye tracking to investigate graph layout effects. In Proceedings of the APVIS 2007. IEEE, 97--100. https://ieeexplore.ieee.org/document/4126225.

[161]

Weidong Huang. 2013. Establishing aesthetics based on human graph reading behavior: Two eye tracking studies. Pers. Ubiquitous Comput. 17, 1 (2013), 93--105.

Digital Library

[162]

Weidong Huang, Peter Eades, and Seok-Hee Hong. 2014. Larger crossing angles make graphs easier to read. J. Vis. Lang. Comput. 25, 4 (2014), 452--465.

Digital Library

[163]

Weidong Huang, Peter Eades, Seok-Hee Hong, and Chun-Cheng Lin. 2013. Improving multiple aesthetics produces better graph drawings. J. Vis. Lang. Comput. 24, 4 (2013), 262--272.

Digital Library

[164]

Weidong Huang, Seok-Hee Hong, and Peter Eades. 2008. Effects of crossing angles. In Proceedings of the PacificVis 2008. IEEE, 41--46. https://ieeexplore.ieee.org/document/4475457.

[165]

Russell Impagliazzo and Ramamohan Paturi. 2001. On the complexity of k-SAT. J. Comput. Syst. Sci. 62, 2 (2001), 367--375.

Digital Library

[166]

Michael Jünger and Petra Mutzel (Eds.). 2004. Graph Drawing Software. Springer.

[167]

Paul C. Kainen. 1974. Some recent results in topological graph theory. In Graphs Combin. (LNCS), Vol. 406. Springer, 76--108. https://link.springer.com/chapter/10.1007/BFb0066436.

[168]

Dmitriy V. Karpov. 2014. An upper bound on the number of edges in an almost planar bipartite graph. J. Math. Sci. 196, 6 (2014), 737--746.

[169]

Michael Kaufmann and Torsten Ueckerdt. 2014. The density of fan-planar graphs. CoRR abs/1403.6184 (2014).

[170]

Michael Kaufmann and Dorothea Wagner (Eds.). 2001. Drawing Graphs. Springer.

[171]

Michael Kaufmann and Roland Wiese. 2002. Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6, 1 (2002), 115--129.

[172]

Ken-ichi Kawarabayashi. 2009. Planarity allowing few error vertices in linear time. In Proceedings of theIEEE FoCS 2009. IEEE, 639--648. https://ieeexplore.ieee.org/document/5438591.

Digital Library

[173]

Ken-ichi Kawarabayashi and Buce Reed. 2007. Computing crossing number in linear time. In Proceedings of the SToC 2007. ACM, 382--390. https://dl.acm.org/citation.cfm?doid=1250790.1250848.

Digital Library

[174]

Philipp Kindermann, Fabrizio Montecchiani, Lena Schlipf, and André Schulz. 2018. Drawing subcubic 1-planar graphs with few bends, few slopes, and large angles. CoRR abs/1808.08496 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_11.

[175]

Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. 2017. An annotated bibliography on 1-planarity. Computer Science Review 25 (2017), 49--67.

[176]

Vladimir P. Korzhik and Bojan Mohar. 2013. Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory 72, 1 (2013), 30--71.

Digital Library

[177]

Yaakov S. Kupitz. 1979. Extremal Problems in Combinatorial Geometry. Matematisk institut, Aarhus Universitet.

[178]

Charles E. Leiserson. 1980. Area-efficient graph layouts (for VLSI). In Proceedings of the FOCS. IEEE Computer Society, 270--281. https://ieeexplore.ieee.org/document/4567828.

Digital Library

[179]

William J. Lenhart, Giuseppe Liotta, and Fabrizio Montecchiani. 2017. On partitioning the edges of 1-plane graphs. Theor. Comput. Sci. 662 (2017), 59--65.

[180]

John M. Lewis and Mihalis Yannakakis. 1980. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20, 2 (1980), 219--230.

[181]

Giuseppe Liotta and Fabrizio Montecchiani. 2016. L-visibility drawings of IC-planar graphs. Inf. Process. Lett. 116, 3 (2016), 217--222.

Digital Library

[182]

Giuseppe Liotta, Fabrizio Montecchiani, and Alessandra Tappini. 2018. Ortho-polygon visibility representations of 3-connected 1-plane graphs. CoRR abs/1807.01247 (2018). Accepted at GD 2018. https://link.springer.com/chapter/10.1007%2F978-3-030-04414-5_37.

[183]

P. C. Liu and R. C. Geldmacher. 1979. On the deletion of nonplanar edges of a graph. In Proceedings of the 10th Southeastern Conference on Combinatorics, Graph Theory, and Computing. 727--738.

[184]

Pierce Mike. 2014. Searching for and classifying the finite set of minor-minimal non-apex graphs. Master’s thesis. California State University.

[185]

Petra Mutzel. 2001. An alternative method to crossing minimization on hierarchical graphs. SIAM J. Optimization 11, 4 (2001), 1065--1080.

Digital Library

[186]

Tomoki Nakamigawa. 2000. A generalization of diagonal flips in a convex polygon. Theor. Comput. Sci. 235, 2 (2000), 271--282.

Digital Library

[187]

Jaroslav Nešetřil, Patrice Ossona de Mendez, and David R. Wood. 2012. Characterisations and examples of graph classes with bounded expansion. Eur. J. Comb. 33, 3 (2012), 350--373.

Digital Library

[188]

Quan Hoang Nguyen, Peter Eades, Seok-Hee Hong, and Weidong Huang. 2011. Large crossing angles in circular layouts. In Proceedings of the GD 2010 (LNCS), Vol. 6502. Springer, 397--399. https://link.springer.com/chapter/10.1007%2F978-3-642-18469-7_40.

[189]

T. A. J. Nicholson. 1968. Permutation procedure for minimising the number of crossings in a network. Proc. IEE 115, 1 (1968), 21--26.

[190]

Taylor L. Ollmann. 1974. On the book thicknesses of various graphs. In Proceedings, 4th S.E. Conference on Combinatorics, Graph Theory, and Computing, Frederick Hoffman, Roy B. Levow, and Robert S. D. Thomas (Eds.). Utilitas Mathematics Publ. Inc., 459.

[191]

János Pach. 2018. Geometric graph theory. In Handbook of Discr. and Comp. Geom., Third Edition. 257--279.

[192]

János Pach, Rom Pinchasi, Micha Sharir, and Géza Tóth. 2005. Topological graphs with no large grids. Graphs Combin. 21, 3 (2005), 355--364.

Digital Library

[193]

János Pach, Rom Pinchasi, Gábor Tardos, and Géza Tóth. 2004. Geometric graphs with no self-intersecting path of length three. Eur. J. Comb. 25, 6 (2004), 793--811.

Digital Library

[194]

János Pach, Rados Raddoicic, Gábor Tardos, and Géza Tóth. 2006. Improving the crossing lemma by finding more crossings in sparse graphs. Discr. Comput. Geom. 36, 4 (2006), 527--552.

Digital Library

[195]

János Pach, Farhad Shahrokhi, and Mario Szegedy. 1996. Applications of the crossing number. Algorithmica 16, 1 (1996), 111--117.

[196]

János Pach and Jenö Töröcsik. 1994. Some geometric applications of dilworth’s theorem. Discr. Comput. Geom. 12 (1994), 1--7.

Digital Library

[197]

János Pach and Géza Tóth. 1997. Graphs drawn with few crossings per edge. Combinatorica 17, 3 (1997), 427--439.

[198]

János Pach and Rephael Wenger. 2001. Embedding planar graphs at fixed vertex locations. Graphs Combin. 17, 4 (2001), 717--728.

[199]

Rom Pinchasi and Rados Raddoicic. 2003. Topological graphs with no self-intersecting cycle of length 4. In Proceedings of the ACM SoCG 2003. ACM, 98--103. https://dl.acm.org/citation.cfm?doid=777792.777807.

Digital Library

[200]

Helen C. Purchase. 2000. Effective information visualisation: A study of graph drawing aesthetics and algorithms. Interacting with Computers 13, 2 (2000), 147--162.

[201]

Helen C. Purchase, David A. Carrington, and Jo-Anne Allder. 2002. Empirical evaluation of aesthetics-based graph layout. Empirical Software Engineering 7, 3 (2002), 233--255.

Digital Library

[202]

Gerhard Ringel. 1965. Ein Sechsfarbenproblem auf der Kugel. Abhandlungen aus dem Mathematischen Seminar der Universitaet Hamburg 29, 1--2 (1965), 107--117.

[203]

Maxwell J. Roberts. 2012. Underground Maps Unravelled: Explorations in Information Design. Maxwell J Roberts.

[204]

Neil Robertson and Paul D. Seymour. 2004. Graph minors. XX. wagner’s conjecture. J. Comb. Theory, Ser. B 92, 2 (2004), 325--357.

Digital Library

[205]

Neil Robertson, Paul D. Seymour, and Robin Thomas. 1993. Hadwiger’s conjecture for K<sub>6</sub>-free graphs. Combinatorica 13, 3 (1993), 279--361.

[206]

Marcus Schaefer. 2013. The graph crossing number and its variants: A survey. Electron. J. Combin. 20 (2013).

[207]

Walter Schnyder. 1990. Embedding planar graphs on the grid. In Proceedings of the ACM-SIAM SoDA 1990. SIAM, 138--148. https://dl.acm.org/citation.cfm?id=320176.320191.

Digital Library

[208]

Thomas C. Shermer. 1996. On rectangle visibility graphs. III. external visibility and complexity. In Proceedings of the CCCG 1996. Carleton University Press, 234--239. http://www.cccg.ca/proceedings/1996/cccg1996_0039.pdf.

Digital Library

[209]

Janet M. Six and Ioannis G. Tollis. 2013. Circular drawing algorithms. In Handbook on Graph Drawing and Visualization., Roberto Tamassia (Ed.). Chapman and Hall/CRC, 285--315.

[210]

S. K. Stein. 1951. Convex maps. In Proc. Am. Math. Soc., Vol. 2. 464--466.

[211]

Kozo Sugiyama, Shojiro Tagawa, and Mitsuhiko Toda. 1981. Methods for visual understanding of hierarchical system structures. IEEE Trans. Systems, Man, and Cybernetics 11, 2 (1981), 109--125.

[212]

Andrew Suk and Bartosz Walczak. 2015. New bounds on the maximum number of edges in k-quasi-planar graphs. Comput. Geom. 50 (2015), 24--33.

Digital Library

[213]

Yusuke Suzuki. 2010. Optimal 1-planar graphs which triangulate other surfaces. Discr. Math. 310, 1 (2010), 6--11.

Digital Library

[214]

Roberto Tamassia. 1987. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16, 3 (1987), 421--444.

Digital Library

[215]

Roberto Tamassia and Ioannis G. Tollis. 1986. A unified approach to visibility representations of planar graphs. Discr. Comput. Geom. 1, 1 (1986), 321--341.

Digital Library

[216]

Carsten Thomassen. 1988. Rectilinear drawings of graphs. J. Graph Theory 12, 3 (1988), 335--341.

[217]

C. D. Toth, J. O’Rourke, and J. E. Goodman. 2017. Handbook of Discrete and Computional Geometry, Third Edition. CRC Press.

[218]

Klaus Truemper. 1992. Matroid Decomposition. Academic Press.

[219]

Leslie G. Valiant. 1981. Universality considerations in VLSI circuits. IEEE Trans. Computers 30, 2 (1981), 135--140.

Digital Library

[220]

Pavel Valtr. 1997. Graph drawings with no pairwise crossing edges. In Proceedings of the GD 1997 (LNCS), Vol. 1353. Springer, 205--218. https://link.springer.com/chapter/10.1007%2F3-540-63938-1_63.

Digital Library

[221]

Pavel Valtr. 1998. On geometric graphs with no pairwise parallel edges. Discr. Comput. Geom. 19, 3 (1998), 461--469.

[222]

Marc J. van Kreveld. 2010. The quality ratio of RAC drawings and planar drawings of planar graphs. In Proceedings of the GD 2010 (LNCS), Vol. 6502. Springer, 371--376. https://link.springer.com/chapter/10.1007%2F978-3-642-18469-7_34.

Digital Library

[223]

Massimo Vignelli. 2008. New York Subway Map, http://secondavenuesagas.com/2008/05/02/mens-vogue-calls-on-vignelli-for-a-long-awaited-update/.

[224]

Klaus Wagner. 1936. Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung 46 (1936), 26--32.

[225]

Colin Ware, Helen C. Purchase, Linda Colpoys, and Matthew McGill. 2002. Cognitive measurements of graph aesthetics. Inf. Vis. 1, 2 (2002), 103--110.

Digital Library

[226]

Martin Wattenberg. 2002. Arc diagrams: Visualizing structure in strings. In Proceedings of the INFOVIS. IEEE, 110--116. https://ieeexplore.ieee.org/document/1173155.

Digital Library

[227]

Stephen K. Wismath. 1985. Characterizing bar line-of-sight graphs. In Proceedings of the ACM SoCG 1985. ACM, 147--152. https://dl.acm.org/citation.cfm?doid=323233.323253.

Digital Library

[228]

Stephen K. Wismath. 1989. Bar-representable visibility graphs and a related network flow problem. https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0051983.

[229]

David R. Wood. 2005. Grid drawings of k-colourable graphs. Comput. Geom. 30, 1 (2005), 25--28.

Digital Library

[230]

Mihalis Yannakakis. 1989. Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38, 1 (1989), 36--67.

Digital Library

[231]

Xin Zhang and Guizhen Liu. 2013. The structure of plane graphs with independent crossings and its applications to coloring problems. Open Math. 11, 2 (2013), 308--321.

[232]

H. Zhou, Panpan Xu, X. Yuan, and H. Qu. 2013. Edge bundling in information visualization. Tsinghua Science and Technology 18, 2 (2013), 145--156.

Cited By

Tiezzi MCiravegna GGori M(2024)Graph Neural Networks for Graph DrawingIEEE Transactions on Neural Networks and Learning Systems10.1109/TNNLS.2022.3184967(1-14)Online publication date: 2024
https://doi.org/10.1109/TNNLS.2022.3184967
Khan BJohnstone MCreighton D(2024)A Many-objective Evolutionary Algorithm Approach for Graph Visualization2024 IEEE International Systems Conference (SysCon)10.1109/SysCon61195.2024.10553494(1-7)Online publication date: 15-Apr-2024
https://doi.org/10.1109/SysCon61195.2024.10553494
Binucci CDi Battista GDidimo WDujmović VHong SKaufmann MLiotta GMorin PTappini A(2024)Graphs Drawn With Some Vertices per Face: Density and RelationshipsIEEE Access10.1109/ACCESS.2024.340107812(68828-68846)Online publication date: 2024
https://doi.org/10.1109/ACCESS.2024.3401078
Show More Cited By

Index Terms

A Survey on Graph Drawing Beyond Planarity
1. Human-centered computing
  1. Visualization
2. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory

Recommendations

Fan-planarity

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt 35], who proved that every n-vertex fan-planar drawing has at most 5 n - 10 edges, and ...
Algorithms and bounds for drawing non-planar graphs with crossing-free subgraphs

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing of G in the plane such that the edges of S are not crossed in by any edge of G__ __ We give positive and ...
On the Planar Split Thickness of Graphs

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest k such that the graph is k-splittable into a planar graph. A k-split operation substitutes a vertex v by at ...

Comments

Information & Contributors

Information

Published In

cover image ACM Computing Surveys

ACM Computing Surveys Volume 52, Issue 1

January 2020

758 pages

ISSN:0360-0300

EISSN:1557-7341

DOI:10.1145/3309872

Editor:
Sartaj Sahni
Department of Computer and Information Science and Engineering

Issue’s Table of Contents

Copyright © 2019 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 21 February 2019

Accepted: 01 October 2018

Revised: 01 September 2018

Received: 01 April 2018

Published in CSUR Volume 52, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Survey
Research
Refereed

Funding Sources

“Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni” - Ricerca di Base 2018, Dip. Eng. Univ. of Perugia

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

68
Total Citations
View Citations
987
Total Downloads

Downloads (Last 12 months)134
Downloads (Last 6 weeks)35

Reflects downloads up to 15 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Tiezzi MCiravegna GGori M(2024)Graph Neural Networks for Graph DrawingIEEE Transactions on Neural Networks and Learning Systems10.1109/TNNLS.2022.3184967(1-14)Online publication date: 2024
https://doi.org/10.1109/TNNLS.2022.3184967
Khan BJohnstone MCreighton D(2024)A Many-objective Evolutionary Algorithm Approach for Graph Visualization2024 IEEE International Systems Conference (SysCon)10.1109/SysCon61195.2024.10553494(1-7)Online publication date: 15-Apr-2024
https://doi.org/10.1109/SysCon61195.2024.10553494
Binucci CDi Battista GDidimo WDujmović VHong SKaufmann MLiotta GMorin PTappini A(2024)Graphs Drawn With Some Vertices per Face: Density and RelationshipsIEEE Access10.1109/ACCESS.2024.340107812(68828-68846)Online publication date: 2024
https://doi.org/10.1109/ACCESS.2024.3401078
Binucci CDi Giacomo ELenhart WLiotta GMontecchiani FNöllenburg MSymvonis A(2024)On the complexity of the storyplan problemJournal of Computer and System Sciences10.1016/j.jcss.2023.103466139:COnline publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1016/j.jcss.2023.103466
Bekos MDa Lozzo GGriesbach SGronemann MMontecchiani FRaftopoulou C(2024)Book embeddings of k-framed graphs and k-map graphsDiscrete Mathematics10.1016/j.disc.2023.113690347:1Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1016/j.disc.2023.113690
Ehlers HVilledieu ARaidou RWu H(2024)Improving readability of static, straight-line graph drawingsComputers and Graphics10.1016/j.cag.2023.09.010116:C(448-463)Online publication date: 4-Mar-2024
https://dl.acm.org/doi/10.1016/j.cag.2023.09.010
Tóth C(2024)Plane Multigraphs with One-Bend and Circular-Arc Edges of a Fixed AngleWALCOM: Algorithms and Computation10.1007/978-981-97-0566-5_3(15-31)Online publication date: 18-Mar-2024
https://dl.acm.org/doi/10.1007/978-981-97-0566-5_3
Antić T(2024)Convex-Geometric k-Planar Graphs Are Convex-Geometric -QuasiplanarCombinatorial Algorithms10.1007/978-3-031-63021-7_11(138-150)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-63021-7_11
Chaplick SKlute FParada IRollin JUeckerdt T(2024)Edge‐minimum saturated k‐planar drawingsJournal of Graph Theory10.1002/jgt.23097106:4(741-762)Online publication date: 28-Mar-2024
https://doi.org/10.1002/jgt.23097
Jindal GKarlapalem K(2023)A Simple yet Useful Spiral Visualization of Large Graphs2023 IEEE Visualization and Visual Analytics (VIS)10.1109/VIS54172.2023.00043(171-175)Online publication date: 21-Oct-2023
https://doi.org/10.1109/VIS54172.2023.00043
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents