Polygon vertex extremality and decomposition of polygons

Abstract

In this paper, we show that if we decompose a polygon into two smaller polygons, then by comparing the number of extremal vertices in the original polygon versus the sum of the two smaller polygons, we can gain at most two globally extremal vertices in the smaller polygons, as well as at most two locally extremal vertices. We then will derive two discrete Four-Vertex Theorems from our results.Definition 2.1. We say that a polygonal curve is generic if the maximal number of vertices that lie on a circle is three and no three vertices are collinear.Observe that all regular polygons are not generic.