### Abstract

Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling. Using a combination of these powerful combinatorial structures, we show that every planar Laman graph has an L-contact representation, that is, planar Laman graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that planar Laman graphs and their subgraphs are the only graphs that can be represented this way. We present efficient algorithms that compute, for every planar Laman graph G, an angular tree, angle labeling, edge labeling, and finally an L-contact representation of G. The overall running time is Script O sign(n ^{2}), where n is the number of vertices of G, and the L-contact representation is realized on the n x n grid.

Original language | English (US) |
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Title of host publication | Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |

Publisher | Association for Computing Machinery |

Pages | 1668-1678 |

Number of pages | 11 |

ISBN (Print) | 9781611972511 |

State | Published - Jan 1 2013 |

Event | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States Duration: Jan 6 2013 → Jan 8 2013 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |
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Country | United States |

City | New Orleans, LA |

Period | 1/6/13 → 1/8/13 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

*Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013*(pp. 1668-1678). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery.