Enumerating non-crossing minimally rigid frameworks

Authors:
David Avis;Naoki Katoh;Makoto Ohsaki;Ileana Streinu;Shin-ichi Tanigawa
Affiliations:
School of Computer Science, McGill University, Canada;Department of Architecture and Architectural Engineering, Kyoto University, Kyoto, Japan;Department of Architecture and Architectural Engineering, Kyoto University, Kyoto, Japan;Dept. of Comp. Science, Smith College, Northampton, MA;Department of Architecture and Architectural Engineering, Kyoto University, Kyoto, Japan
Venue:
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Year:
2006

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Abstract

In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks (simply called non-crossing Laman frameworks) on a given generic set of n points. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all non-crossing Laman frameworks on a given point set is connected by flips which remove an edge and then restore the Laman property with the addition of a non-crossing edge.