On Geometric Dilation and Halving Chords

Abstract

Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of designing graphs of low dilation. We provide a characterization of closed curves of constant halving distance (i.e., curves for which all chords dividing the curve length in half are of constant length) which are useful in this context. We then relate the halving distance of curves to other geometric quantities such as area and width. Among others, this enables us to derive a new upper bound on the geometric dilation of closed curves, as a function of D/w, where D and w are the diameter and width, respectively. We further give lower bounds on the geometric dilation of polygons with n sides as a function of n. Our bounds are tight for centrally symmetric convex polygons.