Enumerating Non-crossing Minimally Rigid Frameworks

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. [PUBLICATION ABSTRACT]