Optimal Reeb graphs on two- and three-connected planar polygon

TL;DR

This paper characterizes all possible optimal Reeb graphs for planar polygons with specific connectivity and extremal points, revealing their topological structures in relation to polygon decomposition.

Contribution

It provides a complete description of optimal Reeb graphs for polygons in general position with one local minimum and maximum, linking them to topological structures of trapezoid maps.

Findings

All possible optimal Reeb graphs for the given polygons are described.

Constructed Reeb graphs correspond to topological structures of specific trapezoid maps.

The study applies to polygons with two- and three-connected planar structures.

Abstract

To investigate the topological structure of planar polygon decomposition on trapezoids, which is formed by height functions. We use the oriented Reeb graph of the function with a marked vertex. We describe all possible optimal Reeb graphs in the case of polygon in general position with one local minimum and one local maximum. By optimal Reeb graph we mean Reeb graph, which cann't be obtaned from other Reeb graph by subdivision of an edge or a leaf attaching. In this case polygon is a triangle with triangle holes. Constructed Reeb graphs give topological structures of trapezoid maps on two-connected polygon with six vertexes and three-connected polygon with nine vertexes.