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In this paper, we show that if we decompose a polygon into two smallerpolygons, then by comparing the number of extremal vertices in the originalpolygon versus the sum of the two smaller polygons, we can gain at most twoglobally extremal vertices in the smaller polygons, as well as at most twolocally extremal vertices. We then will derive two discrete Four-VertexTheorems from our results.

Polygon Vertex Extremality and Decomposition of Polygons