2025/08/01 by Naoki Kitazawa, Kitazawa, Naoki · 3 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Polynomial and algebraic computation #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2508.00729
openalex publication_date 2025/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.