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Embedding Graphs into Embedded Graphs

Author Radoslav Fulek

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Radoslav Fulek

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Radoslav Fulek. Embedding Graphs into Embedded Graphs. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.34

Abstract

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle.
Keywords
  • Graph embedding
  • C-planarity
  • Weakly simple polygons

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References

  1. Hugo A. Akitaya, Greg Aloupis, Jeff Erickson, and Csaba Tóth. Recognizing weakly simple polygons. In 32nd International Symposium on Computational Geometry (SoCG 2016), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:16, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.8.
  2. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, and Fabrizio Frati. Strip planarity testing for embedded planar graphs. Algorithmica, 77(4):1022-1059, 2017. Google Scholar
  3. Patrizio Angelini and Giordano Da Lozzo. Clustered Planarity with Pipes. In Seok-Hee Hong, editor, 27th International Symposium on Algorithms and Computation (ISAAC 2016), volume 64 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:13, 2016. Google Scholar
  4. Therese C. Biedl. Drawing planar partitions III: Two constrained embedding problems. Rutcor Research Report 13-98, 1998. Google Scholar
  5. Thomas Bläsius and Ignaz Rutter. A new perspective on clustered planarity as a combinatorial embedding problem. In Christian A. Duncan and Antonios Symvonis, editors, Graph Drawing - 22nd International Symposium, GD 2014, Würzburg, Germany, September 24-26, 2014, Revised Selected Papers, volume 8871 of Lecture Notes in Computer Science, pages 440-451. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-45803-7_37.
  6. David Dylan Bremner. Point visibility graphs and restricted-orientation polygon covering. PhD thesis, Simon Fraser University, 1993. Google Scholar
  7. Hsien-Chih Chang, Jeff Erickson, and Chao Xu. Detecting weakly simple polygons. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1655-1670, 2015. Google Scholar
  8. Markus Chimani, Giuseppe Di Battista, Fabrizio Frati, and Karsten Klein. Advances on testing c-planarity of embedded flat clustered graphs. In Christian A. Duncan and Antonios Symvonis, editors, Graph Drawing - 22nd International Symposium, GD 2014, Würzburg, Germany, September 24-26, 2014, Revised Selected Papers, volume 8871 of Lecture Notes in Computer Science, pages 416-427. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-45803-7_35.
  9. Pier Francesco Cortese, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Maurizio Pizzonia. C-planarity of c-connected clustered graphs. J. Graph Algorithms Appl., 12(2):225-262, 2008. Google Scholar
  10. Pier Francesco Cortese, Giuseppe Di Battista, Maurizio Patrignani, and Maurizio Pizzonia. On embedding a cycle in a plane graph. Discrete Mathematics, 309(7):1856 - 1869, 2009. Google Scholar
  11. Reinhard Diestel. Graph Theory. Springer, New York, 2010. Google Scholar
  12. István Fáry. On straight line representation of planar graphs. Acta Univ. Szeged. Sect. Sci. Math., 11:229-233, 1948. Google Scholar
  13. Qing-Wen Feng, Robert F. Cohen, and Peter Eades. How to draw a planar clustered graph. In Ding-Zhu Du and Ming Li, editors, Computing and Combinatorics, volume 959 of Lecture Notes in Computer Science, pages 21-30. Springer Berlin Heidelberg, 1995. Google Scholar
  14. Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. In Paul Spirakis, editor, Algorithms — ESA '95, volume 979 of Lecture Notes in Computer Science, pages 213-226. Springer Berlin Heidelberg, 1995. Google Scholar
  15. Radoslav Fulek. C-planarity of embedded cyclic c-graphs. In International Symposium on Graph Drawing and Network Visualization, pages 94-106. Springer, 2016. Google Scholar
  16. Radoslav Fulek and Jan Kyncl. Hanani-tutte for approximating maps of graphs. CoRR, abs/1705.05243, 2017. URL: http://arxiv.org/abs/1705.05243.
  17. Radoslav Fulek, Jan Kynčl, Igor Malinovic, and Dömötör Pálvölgyi. Clustered planarity testing revisited. Electronic Journal of Combinatorics, 22, 2015. Google Scholar
  18. Michael T. Goodrich, George S. Lueker, and Jonathan Z. Sun. C-planarity of extrovert clustered graphs. In Patrick Healy and Nikola S. Nikolov, editors, Graph Drawing, 13th International Symposium, GD 2005, Limerick, Ireland, September 12-14, 2005, Revised Papers, volume 3843 of Lecture Notes in Computer Science, pages 211-222. Springer, 2005. URL: https://doi.org/10.1007/11618058_20, URL: http://dx.doi.org/10.1007/11618058_20.
  19. Carsten Gutwenger, Michael Jünger, Sebastian Leipert, Petra Mutzel, Merijam Percan, and René Weiskircher. Advances in c-planarity testing of clustered graphs. In Stephen G. Kobourov and Michael T. Goodrich, editors, Graph Drawing, 10th International Symposium, GD 2002, Irvine, CA, USA, August 26-28, 2002, Revised Papers, volume 2528 of Lecture Notes in Computer Science, pages 220-235. Springer, 2002. URL: https://doi.org/10.1007/3-540-36151-0_21, URL: http://dx.doi.org/10.1007/3-540-36151-0_21.
  20. Vít Jelínek, Eva Jelínková, Jan Kratochvíl, and Bernard Lidický. Clustered planarity: Embedded clustered graphs with two-component clusters. In Ioannis G. Tollis and Maurizio Patrignani, editors, Graph Drawing, 16th International Symposium, GD 2008, Heraklion, Crete, Greece, September 21-24, 2008. Revised Papers, volume 5417 of Lecture Notes in Computer Science, pages 121-132. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-642-00219-9_13.
  21. Eva Jelínková, Jan Kára, Jan Kratochvíl, Martin Pergel, Ondřej Suchý, and Tomáš Vyskočil. Clustered planarity: Small clusters in cycles and Eulerian graphs. J. Graph Algorithms Appl., 13(3):379-422, 2009. Google Scholar
  22. Piotr Minc. Embedding simplicial arcs into the plane. Topol. Proc. 22, pages 305-340, 1997. Google Scholar
  23. Ares Ribó Mor. Realization and Counting Problems for Planar Structures: Trees and Linkages, Polytopes and Polyominoes. PhD thesis, Freie U., Berlin, 2006. Google Scholar
  24. K Sieklucki. Realization of mappings. Fundamenta Mathematicae, 65(3):325-343, 1969. Google Scholar
  25. Mikhail Skopenkov. On approximability by embeddings of cycles in the plane. Topology and its Applications, 134(1):1-22, 2003. Google Scholar
  26. Godfried Toussaint. On separating two simple polygons by a single translation. Discrete &Computational Geometry, 4(3):265-278, 1989. Google Scholar
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