Description

• Title:
  Realization of a graph as the Reeb graph of a height function on an embedded surface
• Creator:
  Gelbukh, Irina
• Description:
  We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $\mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nd/4.0 ;
  Prawa autorskie (c) 2023 Irina Gelbukh
• Date:
  2023.01.25
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 61, No 2 (June 2023); 591 - 610;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/42231/34564 ;https://apcz.umk.pl/TMNA/article/view/42231/34565
• Subject:
  {Reeb graph;
  height function;
  Morse-Bott function;
  orientable surface;
  embedding;
  Primary: 57R40, 53A05;
  Secondary: 58K05, 57R45
• Identifier: https://apcz.umk.pl/TMNA/article/view/42231
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other