Polygon Vertex Extremality and Decomposition of Polygons

TL;DR

This paper explores how decomposing polygons affects extremal vertices, establishing bounds on extremal vertices in smaller polygons and deriving discrete Four-Vertex Theorems.

Contribution

It introduces bounds on extremal vertices during polygon decomposition and derives two discrete Four-Vertex Theorems from these bounds.

Findings

Decomposition limits extremal vertices to at most two in smaller polygons.

At most two globally extremal vertices can be gained through decomposition.

At most two locally extremal vertices can be introduced in the process.

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.