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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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