Given a plane forest F = (V,E) of |V | = n points, we find the minimum set S ⊆ E of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F.We present an O(n log n)-time algorithm that solves this problem.We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, β-skeleton and Delaunay triangulation.We present an algorithm that identifies the minimum set S ⊆ E of edges of a given plane graph I = (V,E) such that I ⊆ CGβ(V,S) for 1 ≤ β ≤ 2, where CGβ(V,S) is the constraint β-skeleton over the set V of vertices and the set S of constraints.The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given.

Additional Metadata
Keywords Constraints, Delaunay, MST, Proximity graphs, Visibility, β-skeletons
Persistent URL dx.doi.org/10.1007/978-3-319-44543-45
Citation
Bose, P, de Carufel, J.-L. (Jean-Lou), Shaikhet, A. (Alina), & Smid, M. (2016). Essential constraints of edge-constrained proximity graphs. doi:10.1007/978-3-319-44543-45