On the construction of increasing-chord graphs on convex point sets

A geometric path from s to t is increasing-chord, if while traversing it from s to t the distance to the following (resp. from the preceding) points of the path decreases (resp. increases). A geometric graph is increasing-chord if each two distinct vertices are connected with an increasing-chord path. We show that given a convex point set P in the plane we can construct an increasing-chord graph consisting of P, at most one Steiner point and at most 4|P| - 8 edges.