Walter Didimo
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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.
- 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.Google Scholar
Digital Library
- Eyal Ackerman. 2014. A note on 1-planar graphs. Discr. Appl. Math. 175 (2014), 104--108.Google ScholarCross Ref
- Eyal Ackerman. 2015. On topological graphs with at most four crossings per edge. CoRR abs/1509.01932 (2015).Google Scholar
- Eyal Ackerman, Jacob Fox, János Pach, and Andrew Suk. 2014. On grids in topological graphs. Comput. Geom. 47, 7 (2014), 710--723.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Jin Akiyama and Noga Alon. 1989. Disjoint simplices and geometric hypergraphs. Ann. New York Acad. Sci. 555, 1 (1989), 1--3.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Md. Jawaherul Alam, Franz J. Brandenburg, and Stephen G. Kobourov. 2015. On the book thickness of 1-planar graphs. CoRR abs/1510.05891 (2015).Google Scholar
- 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.Google Scholar
- Noga Alon and Paul Erdös. 1989. Disjoint edges in geometric graphs. Discr. Comput. Geom. 4 (1989), 287--290. Google ScholarDigital Library
- 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.Google Scholar
- 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.Google Scholar
- 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.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- Evmorfia N. Argyriou, Michael A. Bekos, and Antonios Symvonis. 2013. Maximizing the total resolution of graphs. Comput. J. 56, 7 (2013), 887--900.Google ScholarCross Ref
- 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.Google Scholar
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- S. Avital and H. Hanani. 1966. Graphs. Gilyonot Lematematika 3 (1966), 2--8.Google Scholar
- Christian Bachmaier, Franz J. Brandenburg, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. 2017. NIC-planar graphs. Discr. Appl. Math. 232 (2017), 23--40. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Michael J. Bannister, Sergio Cabello, and David Eppstein. 2018. Parameterized complexity of 1-planarity. J. Graph Algorithms Appl. 22, 1 (2018), 23--49.Google ScholarCross Ref
- Imre Bárány and Günter Rote. 2005. Strictly convex drawings of planar graphs. CoRR abs/cs/0507030 (2005).Google Scholar
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Carlo Batini, Enrico Nardelli, and Roberto Tamassia. 1986. A layout algorithm for data flow diagrams. IEEE Trans. Software Eng. 12, 4 (1986), 538--546. Google ScholarDigital Library
- Richard A. Becker, Stephen G. Eick, and Allan R. Wilks. 1995. Visualizing network data. IEEE Trans. Vis. Comput. Graph. 1, 1 (1995), 16--28. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google Scholar
- 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.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google Scholar
- 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/.Google Scholar
- 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.Google ScholarCross Ref
- F. Bernhart and P. C. Kainen. 1979. The book thickness of a graph. J. Combin. Theory Ser. B 27 (1979), 320--331.Google ScholarCross Ref
- Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. 2002. Quasi-upward planarity. Algorithmica 32, 3 (2002), 474--506. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google ScholarCross Ref
- Béla Bollobás. 1978. Extremal Graph Theory. Academic Press, New York.Google Scholar
- 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. Google ScholarDigital Library
- Franz Brandenburg. 2018. Recognizing IC-planar and NIC-planar graphs. J. Graph Algorithms Appl. 22, 2 (2018), 239--271.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Franz J. Brandenburg. 2014. 1-visibility representations of 1-planar graphs. J. Graph Algorithms Appl. 18, 3 (2014), 421--438.Google ScholarCross Ref
- Franz J. Brandenburg. 2018. A first order logic definition of beyond-planar graphs. J. Graph Algorithms Appl. 22, 1 (2018), 51--66.Google ScholarCross Ref
- Franz J. Brandenburg. 2018. On fan-crossing and fan-crossing free graphs. Inf. Process. Lett. 138 (2018), 67--71.Google ScholarCross Ref
- Franz J. Brandenburg. 2018. Recognizing optimal 1-planar graphs in linear time. Algorithmica 80, 1 (2018), 1--28. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Peter Brass, Gyula Károlyi, and Pavel Valtr. 2003. A Turán-type extremal theorey of convex geometric graphs. Discr. Comput. Geom. 25 (2003), 275--300.Google ScholarCross Ref
- Peter Braß, William O. J. Moser, and János Pach. 2005. Research Problems in Discrete Geometry. Springer.Google Scholar
- 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.Google ScholarCross Ref
- Till Bruckdorfer and Michael Kaufmann. 2012. Mad at edge crossings? Break the edges!. 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Till Bruckdorfer, Michael Kaufmann, and Fabrizio Montecchiani. 2014. 1-bend orthogonal partial edge drawing. J. Graph Algorithms Appl. 18, 1 (2014), 111--131.Google ScholarCross Ref
- 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.Google Scholar
- Sergio Cabello and Bojan Mohar. 2011. Crossing number and weighted crossing number of near-planar graphs. Algorithmica 60, 3 (2011), 484--504. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google Scholar
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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/.Google Scholar
- 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.Google ScholarDigital Library
- J. Czap and P. Šugerek. 2017. Drawing graph joins in the plane with restrictions on crossings. Filomat 31, 2 (2017), 363--370.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- Hubert de Fraysseix, János Pach, and Richard Pollack. 1990. How to draw a planar graph on a grid. Combinatorica 10, 1 (1990), 41--51.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Erik D. Demaine and Mohammad Taghi Hajiaghayi. 2004. Diameter and treewidth in minor-closed graph families, revisited. Algorithmica 40, 3 (2004), 211--215.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google Scholar
- Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. 1999. Graph Drawing. Prentice Hall, Upper Saddle River, NJ.Google ScholarDigital Library
- Giuseppe Di Battista and Roberto Tamassia. 1988. Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61 (1988), 175--198. Google ScholarDigital Library
- 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.Google ScholarDigital Library
- Emilio Di Giacomo, Walter Didimo, Peter Eades, and Giuseppe Liotta. 2014. 2-layer right angle crossing drawings. Algorithmica 68, 4 (2014), 954--997.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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.Google Scholar
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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.Google Scholar
- Josep Díaz, Jordi Petit, and Maria J. Serna. 2002. A survey of graph layout problems. ACM Comput. Surv. 34, 3 (2002), 313--356. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Walter Didimo. 2013. Density of straight-line 1-planar graph drawings. Inf. Process. Lett. 113, 7 (2013), 236--240. Google ScholarDigital Library
- Walter Didimo. 2016. Upward graph drawing. In Encyclopedia of Algorithms. 2308--2312.Google Scholar
- Walter Didimo, Peter Eades, and Giuseppe Liotta. 2010. A characterization of complete bipartite RAC graphs. Inf. Process. Lett. 110, 16 (2010), 687--691. Google ScholarDigital Library
- Walter Didimo, Peter Eades, and Giuseppe Liotta. 2011. Drawing graphs with right angle crossings. Theor. Comput. Sci. 412, 39 (2011), 5156--5166.Google ScholarCross Ref
- 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.Google Scholar
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Walter Didimo and Fabrizio Montecchiani. 2014. Fast layout computation of clustered networks: Algorithmic advances and experimental analysis. Inf. Sci. 260 (2014), 185--199. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Vida Dujmovic. 2015. Graph layouts via layered separators. J. Comb. Theory, Ser. B 110 (2015), 79--89. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Vida Dujmovic and Fabrizio Frati. 2018. Stack and queue layouts via layered separators. J. Graph Algorithms Appl. 22, 1 (2018), 89--99.Google ScholarCross Ref
- Vida Dujmovic, Joachim Gudmundsson, Pat Morin, and Thomas Wolle. 2011. Notes on large angle crossing graphs. Chicago J. Theor. Comput. Sci. 2011 (2011).Google Scholar
- 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.Google ScholarCross Ref
- Vida Dujmovic and David R. Wood. 2004. On linear layouts of graphs. Discr. Math. 8 Theor. Comp. Sci. 6, 2 (2004), 339--358.Google Scholar
- 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.Google Scholar
- 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.Google ScholarCross Ref
- Peter Eades and Giuseppe Liotta. 2013. Right angle crossing graphs and 1-planarity. Discr. Appl. Math. 161, 7--8 (2013), 961--969. Google ScholarDigital Library
- 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/∼bdm/papers/EdgeCrossing.pdf.Google Scholar
- Peter Eades and Nicholas C. Wormald. 1994. Edge crossings in drawings of bipartite graphs. Algorithmica 11, 4 (1994), 379--403. Google ScholarDigital Library
- David Eppstein. 2000. Diameter and treewidth in minor-closed graph families. Algorithmica 27, 3 (2000), 275--291.Google ScholarCross Ref
- David Eppstein, Michael T. Goodrich, and Jeremy Yu Meng. 2007. Confluent layered drawings. Algorithmica 47, 4 (2007), 439--452.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- István Fáry. 1948. On straight-line representation of planar graphs. Acta Sci. Math. 11 (1948), 229--233.Google Scholar
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- M. R. Garey and D. S. Johnson. 1983. Crossing number is NP-complete. SIAM J. Alg. Discr. Meth. 4 (1983), 312--316.Google ScholarDigital Library
- Ashim Garg and Roberto Tamassia. 2001. On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31, 2 (2001), 601--625. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Alexander Grigoriev and Hans L. Bodlaender. 2007. Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49, 1 (2007), 1--11. Google ScholarDigital Library
- Luca Grilli. 2018. On the NP-hardness of GRacSim drawing and k-SEFE Problems. J. Graph Algorithms Appl. 22, 1 (2018), 101--116.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Carsten Gutwenger, Petra Mutzel, and René Weiskircher. 2005. Inserting an edge into a planar graph. Algorithmica 41, 4 (2005), 289--308.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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/.Google Scholar
- Danny Holten. 2006. Hierarchical edge bundles: Visualization of adjacency relations in hierarchical data. IEEE Trans. Vis. Comput. Graph. 12, 5 (2006), 741--748. Google ScholarDigital Library
- Danny Holten and Jarke J. van Wijk. 2009. Force-directed edge bundling for graph visualization. Comput. Graph. Forum 28, 3 (2009), 983--990. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- John E. Hopcroft and Robert Endre Tarjan. 1974. Efficient planarity testing. J. ACM 21, 4 (1974), 549--568. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Weidong Huang. 2013. Establishing aesthetics based on human graph reading behavior: Two eye tracking studies. Pers. Ubiquitous Comput. 17, 1 (2013), 93--105. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Russell Impagliazzo and Ramamohan Paturi. 2001. On the complexity of k-SAT. J. Comput. Syst. Sci. 62, 2 (2001), 367--375. Google ScholarDigital Library
- Michael Jünger and Petra Mutzel (Eds.). 2004. Graph Drawing Software. Springer.Google Scholar
- 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.Google Scholar
- 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.Google ScholarCross Ref
- Michael Kaufmann and Torsten Ueckerdt. 2014. The density of fan-planar graphs. CoRR abs/1403.6184 (2014).Google Scholar
- Michael Kaufmann and Dorothea Wagner (Eds.). 2001. Drawing Graphs. Springer.Google Scholar
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google Scholar
- Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. 2017. An annotated bibliography on 1-planarity. Computer Science Review 25 (2017), 49--67.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Yaakov S. Kupitz. 1979. Extremal Problems in Combinatorial Geometry. Matematisk institut, Aarhus Universitet.Google Scholar
- 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. Google ScholarDigital Library
- William J. Lenhart, Giuseppe Liotta, and Fabrizio Montecchiani. 2017. On partitioning the edges of 1-plane graphs. Theor. Comput. Sci. 662 (2017), 59--65.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- Giuseppe Liotta and Fabrizio Montecchiani. 2016. L-visibility drawings of IC-planar graphs. Inf. Process. Lett. 116, 3 (2016), 217--222. Google ScholarDigital Library
- 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.Google Scholar
- 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.Google Scholar
- Pierce Mike. 2014. Searching for and classifying the finite set of minor-minimal non-apex graphs. Master’s thesis. California State University.Google Scholar
- Petra Mutzel. 2001. An alternative method to crossing minimization on hierarchical graphs. SIAM J. Optimization 11, 4 (2001), 1065--1080. Google ScholarDigital Library
- Tomoki Nakamigawa. 2000. A generalization of diagonal flips in a convex polygon. Theor. Comput. Sci. 235, 2 (2000), 271--282. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- T. A. J. Nicholson. 1968. Permutation procedure for minimising the number of crossings in a network. Proc. IEE 115, 1 (1968), 21--26.Google Scholar
- 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.Google Scholar
- János Pach. 2018. Geometric graph theory. In Handbook of Discr. and Comp. Geom., Third Edition. 257--279.Google Scholar
- János Pach, Rom Pinchasi, Micha Sharir, and Géza Tóth. 2005. Topological graphs with no large grids. Graphs Combin. 21, 3 (2005), 355--364. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarDigital Library
- János Pach, Farhad Shahrokhi, and Mario Szegedy. 1996. Applications of the crossing number. Algorithmica 16, 1 (1996), 111--117.Google ScholarCross Ref
- János Pach and Jenö Töröcsik. 1994. Some geometric applications of dilworth’s theorem. Discr. Comput. Geom. 12 (1994), 1--7. Google ScholarDigital Library
- János Pach and Géza Tóth. 1997. Graphs drawn with few crossings per edge. Combinatorica 17, 3 (1997), 427--439.Google ScholarCross Ref
- János Pach and Rephael Wenger. 2001. Embedding planar graphs at fixed vertex locations. Graphs Combin. 17, 4 (2001), 717--728.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Helen C. Purchase. 2000. Effective information visualisation: A study of graph drawing aesthetics and algorithms. Interacting with Computers 13, 2 (2000), 147--162.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Gerhard Ringel. 1965. Ein Sechsfarbenproblem auf der Kugel. Abhandlungen aus dem Mathematischen Seminar der Universitaet Hamburg 29, 1--2 (1965), 107--117.Google Scholar
- Maxwell J. Roberts. 2012. Underground Maps Unravelled: Explorations in Information Design. Maxwell J Roberts.Google Scholar
- Neil Robertson and Paul D. Seymour. 2004. Graph minors. XX. wagner’s conjecture. J. Comb. Theory, Ser. B 92, 2 (2004), 325--357. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Marcus Schaefer. 2013. The graph crossing number and its variants: A survey. Electron. J. Combin. 20 (2013).Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google Scholar
- S. K. Stein. 1951. Convex maps. In Proc. Am. Math. Soc., Vol. 2. 464--466.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Yusuke Suzuki. 2010. Optimal 1-planar graphs which triangulate other surfaces. Discr. Math. 310, 1 (2010), 6--11. Google ScholarDigital Library
- Roberto Tamassia. 1987. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16, 3 (1987), 421--444. Google ScholarDigital Library
- Roberto Tamassia and Ioannis G. Tollis. 1986. A unified approach to visibility representations of planar graphs. Discr. Comput. Geom. 1, 1 (1986), 321--341. Google ScholarDigital Library
- Carsten Thomassen. 1988. Rectilinear drawings of graphs. J. Graph Theory 12, 3 (1988), 335--341.Google ScholarCross Ref
- C. D. Toth, J. O’Rourke, and J. E. Goodman. 2017. Handbook of Discrete and Computional Geometry, Third Edition. CRC Press.Google Scholar
- Klaus Truemper. 1992. Matroid Decomposition. Academic Press.Google Scholar
- Leslie G. Valiant. 1981. Universality considerations in VLSI circuits. IEEE Trans. Computers 30, 2 (1981), 135--140. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Pavel Valtr. 1998. On geometric graphs with no pairwise parallel edges. Discr. Comput. Geom. 19, 3 (1998), 461--469.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- Massimo Vignelli. 2008. New York Subway Map, http://secondavenuesagas.com/2008/05/02/mens-vogue-calls-on-vignelli-for-a-long-awaited-update/.Google Scholar
- Klaus Wagner. 1936. Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung 46 (1936), 26--32.Google Scholar
- Colin Ware, Helen C. Purchase, Linda Colpoys, and Matthew McGill. 2002. Cognitive measurements of graph aesthetics. Inf. Vis. 1, 2 (2002), 103--110. Google ScholarDigital Library
- Martin Wattenberg. 2002. Arc diagrams: Visualizing structure in strings. In Proceedings of the INFOVIS. IEEE, 110--116. https://ieeexplore.ieee.org/document/1173155. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google Scholar
- David R. Wood. 2005. Grid drawings of k-colourable graphs. Comput. Geom. 30, 1 (2005), 25--28. Google ScholarDigital Library
- Mihalis Yannakakis. 1989. Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38, 1 (1989), 36--67. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- H. Zhou, Panpan Xu, X. Yuan, and H. Qu. 2013. Edge bundling in information visualization. Tsinghua Science and Technology 18, 2 (2013), 145--156.Google ScholarCross Ref
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