Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions

TL;DR

This paper explores how certain graphs, especially those decomposed into trees, can be realized as Reeb graphs of explicit real algebraic functions, linking graph theory with algebraic geometry.

Contribution

It introduces methods to realize graphs with tree decompositions as Reeb graphs of real algebraic functions, expanding understanding of their geometric and topological properties.

Findings

Reeb graphs of height functions on spheres are simple graphs with one or two edges.

Graphs decomposed into trees can be realized as Reeb graphs of algebraic functions.

The paper provides explicit constructions for such realizations.

Abstract

We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of all components of preimages of all single points. Reeb graphs have been fundamental and strong tools in geometry of manifolds since the birth of theory of Morse functions, in the former half of the 20th century. We can easily see that the Reeb graph of the natural height of the unit sphere whose dimension is at least $2$ is a graph with exactly one edge and two edges. We are concerned with realizations of graphs decomposed into trees nicely, each vertex of which corresponds to a graph with exactly one edge and two edges or a graph with exactly two edges homeomorphic to a circle.