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Rigid Two-Dimensional Frameworks with Two Coincident Points

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Abstract

Let G = (V, E) be a graph and \({u, v \in V}\) be two distinct vertices. We give a necessary and sufficient condition for the existence of an infinitesimally rigid two-dimensional bar-and-joint framework (G, p), in which the positions of u and v coincide. We also determine the rank function of the corresponding modified generic rigidity matroid on ground-set E. The results lead to efficient algorithms for testing whether a graph has such a coincident realization with respect to a designated vertex pair and, more generally, for computing the rank of G in the matroid.

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Authors and Affiliations

  1. Data Mining and Web Search Research Group, Informatics Laboratory, Computer and Automation Research Institute, Hungarian Academy of Sciences, Kende u. 13-17, 1111, Budapest, Hungary

    Zsolt Fekete

  2. Department of Operations Research, Eötvös University, Pázmány sétány 1/C, 1117, Budapest, Hungary

    Tibor Jordán & Viktória E. Kaszanitzky

  3. MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Budapest, Hungary

    Tibor Jordán

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  1. Zsolt Fekete
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  2. Tibor Jordán
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  3. Viktória E. Kaszanitzky
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Correspondence to Tibor Jordán.

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Fekete, Z., Jordán, T. & Kaszanitzky, V.E. Rigid Two-Dimensional Frameworks with Two Coincident Points. Graphs and Combinatorics 31, 585–599 (2015). https://doi.org/10.1007/s00373-013-1390-0

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  • DOI: https://doi.org/10.1007/s00373-013-1390-0

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