Enumerating Constrained Non-crossing Geometric Spanning Trees

In this paper we present an algorithm for enumerating without repetitions all non-crossing geometric spanning trees on a given set of n points in the plane under edge constraints (i.e., some edges are required to be included in spanning trees). We will first prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the constrained smallest indexed triangulation which is obtained by extending the lexicographically ordered triangulation introduced by Bespamyatnikh. More specifically, we prove that all edge-constrained triangulations can be transformed to the smallest indexed triangulation among them by O(n) times of greedy flips. Our enumeration algorithm is based on the reverse search paradigm of Avis and Fukuda, and it generates each output graph in O(n) time and O(n) space. This result improves the previous O(n) bound by Avis and Fukuda for the unconstrained case by factor of O(n). For the edge constrained case, the previous algorithm cannot be extended so as to cope with edge constraints. However, our algorithm can deal with the edge-constrained case in the same running time.