Planar graphs embedded in generic ways and realizing them as Reeb graphs of real algebraic functions

TL;DR

This paper demonstrates that generically embedded planar graphs can be realized as Reeb graphs of real algebraic functions constructed from elementary polynomials, advancing the understanding of the topological representation of such graphs.

Contribution

The paper provides a new method to realize planar graphs as Reeb graphs of real algebraic functions using elementary polynomials and procedures, showing generic embeddability.

Findings

Planar graphs can be realized as Reeb graphs of algebraic functions.

Real algebraic functions can be constructed from elementary polynomials.

Generically embedded planar graphs are homeomorphic to Reeb graphs.

Abstract

This paper is concerned with long-time interest of us, especially, the author, in realizing graphs as Reeb graphs of real algebraic functions of certain nice classes. The Reeb graph of a differentiable function is the set consisting of all components of preimages of all single points and endowed with the quotient topology canonically. In tame cases, such objects are graphs. The Reeb graph of the natural height of the unit sphere of dimension at least $2$ is a graph with exactly one edge and homeomorphic to a closed interval. These graphs have been fundamental and strong tools in geometry since theory of Morse functions has been established in the former half of the last century. We present a new answer to the problem, saying that generically embedded planar graphs are homeomorphic to the Reeb graphs of real algebraic functions obtained by elementary polynomials and elementary procedures.