In this paper, we present algorithms for enumerating, without repetitions, all triangulations and non-crossing geometric spanning trees on a given set of n points in the plane under edge inclusion constraint (i.e., some edges are required to be included in the graph). We will first extend the lexicographically ordered triangulations introduced by Bespamyatnikh to the edge-constrained case, and then we prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the edge-constrained lexicographically largest triangulation. More specifically, we prove that all edge-constrained triangulations can be transformed to the lexicographically largest triangulation among them by O(n2) greedy flips, i.e., by greedily increasing the lexicographical ordering of the edge list, and a similar result also holds for a set of edge-constrained non-crossing spanning trees. Our enumeration algorithms generate each output triangulation and non-crossing spanning tree in O(loglogn) and O(n2) time, respectively, based on the reverse search technique.
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