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Recognizing Planar Laman Graphs

Authors Jonathan Rollin, Lena Schlipf, André Schulz



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Jonathan Rollin
  • Theoretische Informatik, FernUniversität in Hagen, Hagen, Germany
Lena Schlipf
  • Theoretische Informatik, FernUniversität in Hagen, Hagen, Germany
André Schulz
  • Theoretische Informatik, FernUniversität in Hagen, Hagen, Germany

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Jonathan Rollin, Lena Schlipf, and André Schulz. Recognizing Planar Laman Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 79:1-79:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.79

Abstract

Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}). To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • planar graphs
  • Laman graphs
  • network flow
  • connectivity

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References

  1. Nieke Aerts and Stefan Felsner. Straight line triangle representations. Discrete Comput. Geom., 57(2):257-280, 2017. Google Scholar
  2. Jørgen Bang-Jensen and Gregory Gutin. Digraphs. Springer Monographs in Mathematics. Springer, London, second edition, 2009. Google Scholar
  3. Alex R. Berg and Tibor Jordán. Algorithms for graph rigidity and scene analysis. In Algorithms - ESA 2003, volume 2832 of Lecture Notes in Comput. Sci., pages 78-89. Springer, Berlin, 2003. Google Scholar
  4. Glencora Borradaile, David Eppstein, Amir Nayyeri, and Christian Wulff-Nilsen. All-pairs minimum cuts in near-linear time for surface-embedded graphs. In 32nd International Symposium on Computational Geometry (SoCG'16), volume 51 of LIPIcs. Leibniz Int. Proc. Inform., pages 22:1-22:16, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. Google Scholar
  5. Glencora Borradaile, Philip N. Klein, Shay Mozes, Yahav Nussbaum, and Christian Wulff-Nilsen. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. SIAM J. Comput., 46(4):1280-1303, 2017. Google Scholar
  6. Robert Connelly, Erik D. Demaine, and Günter Rote. Straightening Polygonal Arcs and Convexifying Polygonal Cycles. Discrete Comput. Geom., 30(2):205-239, 2003. Google Scholar
  7. Ovidiu Daescu and Anastasia Kurdia. Towards an optimal algorithm for recognizing Laman graphs. J. Graph Algorithms Appl., 13(2):219-232, 2009. Google Scholar
  8. David Eisenstat and Philip N. Klein. Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs. In 45th ACM Symposium on Theory of Computing (STOC'13), pages 735-744. ACM, New York, 2013. Google Scholar
  9. Zsolt Fekete, Tibor Jordán, and Walter Whiteley. An inductive construction for plane Laman graphs via vertex splitting. In Algorithms - ESA 2004, volume 3221 of Lecture Notes in Comput. Sci., pages 299-310. Springer, Berlin, 2004. Google Scholar
  10. Lester R. Ford, Jr. and Delbert R. Fulkerson. Flows in networks. Princeton University Press, Princeton, N.J., 1962. Google Scholar
  11. Harold N. Gabow. A matroid approach to finding edge connectivity and packing arborescences. J. Comput. System Sci., 50(2):259-273, 1995. Google Scholar
  12. Harold N. Gabow. Using expander graphs to find vertex connectivity. J. ACM, 53(5):800-844, 2006. Google Scholar
  13. Harold N. Gabow and Herbert H. Westermann. Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica, 7(5-6):465-497, 1992. Google Scholar
  14. Loukas Georgiadis. Testing 2-vertex connectivity and computing pairs of vertex-disjoint s-t paths in digraphs. In 37th International Colloquium on Automata, Languages and Programming (ICALP'10), volume 6198 of Lecture Notes in Comput. Sci., pages 738-749. Springer, Berlin, 2010. Google Scholar
  15. Loukas Georgiadis, Daniel Graf, Giuseppe F. Italiano, Nikos Parotsidis, and Przemysław Uznański. All-pairs 2-reachability in O(n^ωlog n) time. In 44th International Colloquium on Automata, Languages, and Programming, volume 80 of LIPIcs. Leibniz Int. Proc. Inform., pages 74:1-74:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. Google Scholar
  16. Ruth Haas, David Orden, Günter Rote, Francisco Santos, Brigitte Servatius, Herman Servatius, Diane Souvaine, Ileana Streinu, and Walter Whiteley. Planar minimally rigid graphs and pseudo-triangulations. Comput. Geom., 31(1-2):31-61, 2005. Google Scholar
  17. Lebrecht Henneberg. Die Graphische Statik der Starren Körper. In Felix Klein and Conrad Müller, editors, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Vierter Band: Mechanik, pages 345-434. Vieweg+Teubner Verlag, Wiesbaden, 1908. Google Scholar
  18. Jacob Holm, Eva Rotenberg, and Mikkel Thorup. Planar reachability in linear space and constant time. In 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS'15), pages 370-389. IEEE Computer Society, Los Alamitos, CA, 2015. Google Scholar
  19. Giuseppe F. Italiano, Luigi Laura, and Federico Santaroni. Finding strong bridges and strong articulation points in linear time. Theoret. Comput. Sci., 447:74-84, 2012. Google Scholar
  20. Donald J. Jacobs and Bruce Hendrickson. An algorithm for two-dimensional rigidity percolation: the pebble game. J. Comput. Phys., 137(2):346-365, 1997. Google Scholar
  21. Tibor Jordán. Combinatorial rigidity: graphs and matroids in the theory of rigid frameworks. In Discrete geometric analysis, volume 34 of MSJ Mem., pages 33-112. Math. Soc. Japan, Tokyo, 2016. Google Scholar
  22. Haim Kaplan and Yahav Nussbaum. Maximum flow in directed planar graphs with vertex capacities. Algorithmica, 61(1):174-189, 2011. Google Scholar
  23. Stephen Kobourov, Torsten Ueckerdt, and Kevin Verbeek. Combinatorial and geometric properties of planar Laman graphs. In 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'12), pages 1668-1678. SIAM, Philadelphia, PA, 2012. Google Scholar
  24. Jakub Ła̧cki, Yahav Nussbaum, Piotr Sankowski, and Christian Wulff-Nilsen. Single Source - All Sinks Max Flows in Planar Digraphs. In 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS'12), pages 599-608. IEEE Computer Society, Washington, DC, 2012. Google Scholar
  25. Gerard Laman. On graphs and rigidity of plane skeletal structures. J. Engrg. Math., 4:331-340, 1970. Google Scholar
  26. Audrey Lee and Ileana Streinu. Pebble game algorithms and sparse graphs. Discrete Math., 308(8):1425-1437, 2008. Google Scholar
  27. Richard J. Lipton and Robert E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math., 36(2):177-189, 1979. Google Scholar
  28. Shay Mozes, Kirill Nikolaev, Yahav Nussbaum, and Oren Weimann. Minimum cut of directed planar graphs in O(nlog log n) time. In 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'18), pages 477-494. SIAM, Philadelphia, PA, 2018. Google Scholar
  29. David Orden, Francisco Santos, Brigitte Servatius, and Herman Servatius. Combinatorial pseudo-triangulations. Discrete Math., 307(3-5):554-566, 2007. Google Scholar
  30. James B. Orlin. Max flows in O(nm) time, or better. In 24th ACM Symposium on Theory of Computing (STOC'13), pages 765-774. ACM, New York, 2013. Google Scholar
  31. Günter Rote, Francisco Santos, and Ileana Streinu. Pseudo-triangulations - a survey. In Surveys on discrete and computational geometry, volume 453 of Contemp. Math., pages 343-410. Amer. Math. Soc., Providence, RI, 2008. Google Scholar
  32. Jens M. Schmidt. A simple test on 2-vertex- and 2-edge-connectivity. Inform. Process. Lett., 113(7):241-244, 2013. Google Scholar
  33. Ileana Streinu. Pseudo-triangulations, rigidity and motion planning. Discrete Comput. Geom., 34(4):587-635, 2005. Google Scholar
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