Increasing-Chord Graphs On Point Sets

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. The main intuition behind this result is that Gabriel triangulations are increasing-chord graphs, a fact which might be of independent interest. Further, we prove that, for every convex point set P with n points, there exists an increasing-chord graph with O(n log n) edges (and with no Steiner points) spanning P.