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Toward Grünbaum’s Conjecture for 4-Connected Graphs

Author Christian Ortlieb

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Author Details

Christian Ortlieb
  • Institute of Computer Science, University of Rostock, Germany

Funding

This research is supported by the grant SCHM 3186/2-1 (401348462) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

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Christian Ortlieb. Toward Grünbaum’s Conjecture for 4-Connected Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.77

Abstract

Given a spanning tree T of a 3-connected planar graph G, the co-tree of T is the spanning tree of the dual graph G^* given by the duals of the edges that are not in T. Grünbaum conjectured in 1970 that there is such a spanning tree T such that T and its co-tree both have maximum degree at most 3. In 2014, Biedl proved that there is a spanning tree T such that T and its co-tree have maximum degree at most 5. Using structural insights into Schnyder woods, Schmidt and the author recently improved this bound on the maximum degree to 4. In this paper, we prove that in a 4-connected planar graph there exists a spanning tree T of maximum degree at most 3 such its co-tree has maximum degree at most 4. This almost solves Grünbaum’s conjecture for 4-connected graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • 4-connected planar graph
  • spanning tree
  • maximum degree
  • Schnyder wood
  • Grünbaum

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References

  1. Md. J. Alam, W. Evans, S. G. Kobourov, S. Pupyrev, J. Toeniskoetter, and T. Ueckerdt. Contact representations of graphs in 3D. In Proceedings of the 14th International Symposium on Algorithms and Data Structures (WADS '15), volume 9214 of Lecture Notes in Computer Science, pages 14-27, 2015. Technical Report accessible on arXiv: https://arxiv.org/abs/1501.00304. URL: https://doi.org/10.1007/978-3-319-21840-3_2.
  2. M. Badent, U. Brandes, and S. Cornelsen. More canonical ordering. Journal of Graph Algorithms and Applications, 15(1):97-126, 2011. URL: https://doi.org/10.7155/JGAA.00219.
  3. D. Barnette. Trees in polyhedral graphs. Canadian Journal of Mathematics, 18:731-736, 1966. URL: https://doi.org/10.4153/CJM-1966-073-4.
  4. T. Biedl. Trees and co-trees with bounded degrees in planar 3-connected graphs. In 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT'14), pages 62-73, 2014. URL: https://doi.org/10.1007/978-3-319-08404-6_6.
  5. Nicolas Bonichon, Stefan Felsner, and Mohamed Mosbah. Convex drawings of 3-connected plane graphs. Algorithmica, 47(4):399-420, 2007. Google Scholar
  6. Luca Castelli Aleardi and Olivier Devillers. Array-based compact data structures for triangulations: practical solutions with theoretical guarantees. J. Comput. Geom., 9(1):247-289, 2018. URL: https://doi.org/10.20382/jocg.v9i1a8.
  7. Julien Courtiel, Eric Fusy, Mathias Lepoutre, and Marni Mishna. Bijections for Weyl chamber walks ending on an axis, using arc diagrams and Schnyder woods. European J. Combin., 69:126-142, 2018. URL: https://doi.org/10.1016/j.ejc.2017.10.003.
  8. P. O. de Mendez. Orientations bipolaires. PhD thesis, École des Hautes Études en Sciences Sociales, Paris, 1994. Google Scholar
  9. G. Di Battista, R. Tamassia, and L. Vismara. Output-sensitive reporting of disjoint paths. Algorithmica, 23(4):302-340, 1999. URL: https://doi.org/10.1007/PL00009264.
  10. Emilio Di Giacomo, Giuseppe Liotta, and Tamara Mchedlidze. Lower and upper bounds for long induced paths in 3-connected planar graphs. Theoret. Comput. Sci., 636:47-55, 2016. URL: https://doi.org/10.1016/j.tcs.2016.04.034.
  11. R. Diestel. Graph theory. Graduate texts in mathematics 173. Springer, Berlin, 4th edition edition, 2012. URL: http://swbplus.bsz-bw.de/bsz377230375cov.htm.
  12. S. Felsner. Geodesic embeddings and planar graphs. Order, 20:135-150, 2003. Google Scholar
  13. S. Felsner. Geometric Graphs and Arrangements. Advanced Lectures in Mathematics. Vieweg+Teubner, Wiesbaden, 2004. URL: https://doi.org/10.1007/978-3-322-80303-0.
  14. S. Felsner. Lattice structures from planar graphs. Electronic Journal of Combinatorics, 11(1):R15, 1-24, 2004. URL: https://doi.org/10.37236/1768.
  15. Stefan Felsner. Convex drawings of planar graphs and the order dimension of 3-polytopes. Order, 18(1):19-37, 2001. URL: https://doi.org/10.1023/A:1010604726900.
  16. B. Grünbaum. Polytopes, graphs, and complexes. Bulletin of the American Mathematical Society, 76(6):1131-1201, 1970. URL: https://doi.org/10.1090/S0002-9904-1970-12601-5.
  17. Xin He and Huaming Zhang. A simple routing algorithm based on Schnyder coordinates. Theoret. Comput. Sci., 494:112-121, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.017.
  18. G. Kant. Drawing planar graphs using the lmc-ordering. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science (FOCS'92), pages 101-110, 1992. URL: https://doi.org/10.1109/SFCS.1992.267814.
  19. G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16(1):4-32, 1996. URL: https://doi.org/10.1007/BF02086606.
  20. Christian Ortlieb and Jens M. Schmidt. Toward Grünbaum’s Conjecture. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024), volume 294 of Leibniz International Proceedings in Informatics (LIPIcs), pages 37:1-37:17, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SWAT.2024.37.
  21. Walter Schnyder. Embedding planar graphs on the grid. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '90, pages 138-148, USA, 1990. Society for Industrial and Applied Mathematics. Google Scholar
  22. Hendrik Schrezenmaier. Homothetic triangle contact representations. In Graph-theoretic concepts in computer science, volume 10520 of Lecture Notes in Comput. Sci., pages 425-437. Springer, Cham, 2017. URL: https://doi.org/10.1007/978-3-319-68705-6_3.
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