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Regions surrounded by circles whose Poincaré-Reeb graphs are trees

Naoki Kitazawa

TL;DR

This work analyzes regions in the plane bounded by circles through the lens of Poincaré-Reeb graphs, focusing on when such graphs form trees. It introduces two local circle-adding operations, MBCC addition and SSCC addition, to iteratively build SS-regions from a disk and tracks the induced changes in the Poincaré-Reeb graph, formalizing a constructive inductive framework. The main result provides a revised, explicit description showing that repeated MBCC/SSCC additions yield a tree structure, with a detailed graph-theoretic evolution via graphs $G_j$ and a correspondence between vertex types, ensuring the tree property. The paper also connects these geometric procedures to moment-map-like real algebraic maps, via a moment-like construction and a Reeb-graph isomorphism, suggesting practical methods to realize prescribed planar regions as algebraic images. Overall, the results offer a concrete, constructive pathway for realizing and manipulating planar regions with prescribed topological and combinatorial structures, with implications for explicit real algebraic mapping problems.

Abstract

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincaré-Reeb graphs are trees and investigate the trees obtained by a certain inductive rule from a disk in the plane. The Poincaré-Reeb graph of such a region is a graph whose underlying set is the set of all components of level sets of the restriction of the canonical projection to the closure and whose vertices are points corresponding to the components containing {\it singular} points. Related studies were started by the author, motivated by importance and difficulty of explicit construction of a real algebraic map onto a prescribed closed region in the plane.

Regions surrounded by circles whose Poincaré-Reeb graphs are trees

TL;DR

This work analyzes regions in the plane bounded by circles through the lens of Poincaré-Reeb graphs, focusing on when such graphs form trees. It introduces two local circle-adding operations, MBCC addition and SSCC addition, to iteratively build SS-regions from a disk and tracks the induced changes in the Poincaré-Reeb graph, formalizing a constructive inductive framework. The main result provides a revised, explicit description showing that repeated MBCC/SSCC additions yield a tree structure, with a detailed graph-theoretic evolution via graphs and a correspondence between vertex types, ensuring the tree property. The paper also connects these geometric procedures to moment-map-like real algebraic maps, via a moment-like construction and a Reeb-graph isomorphism, suggesting practical methods to realize prescribed planar regions as algebraic images. Overall, the results offer a concrete, constructive pathway for realizing and manipulating planar regions with prescribed topological and combinatorial structures, with implications for explicit real algebraic mapping problems.

Abstract

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincaré-Reeb graphs are trees and investigate the trees obtained by a certain inductive rule from a disk in the plane. The Poincaré-Reeb graph of such a region is a graph whose underlying set is the set of all components of level sets of the restriction of the canonical projection to the closure and whose vertices are points corresponding to the components containing {\it singular} points. Related studies were started by the author, motivated by importance and difficulty of explicit construction of a real algebraic map onto a prescribed closed region in the plane.

Paper Structure

This paper contains 10 sections, 2 theorems, 3 figures.

Table of Contents

  1. Introduction.
  2. Fundamental terminologies and notation.
  3. An SS-region: a region surrounded by circles
  4. The content of the present paper and our main result.
  5. On Main Theorem \ref{['mthm:1']}.
  6. Exposition on two kinds of operations adding circles and changing SS-regions.
  7. Adding a small circle centered at a point in the boundary of $D$.
  8. Adding a circle passing two distinct and sufficiently close points in the boundary of $D$.
  9. Main Theorem \ref{['mthm:1']} revisited in an improved and revised form.
  10. Additional exposition on a moment-like map whose image is the closure of the region $D$ for an SS-region $(D,\{S_j\}_{j=1}^l)$.

Key Result

Corollary 1

If we do an MBCC addition to an SS-region $(D,\{S_j\}_{j=1}^l)$, then the Poincaré-Reeb graph of $(D,\{S_j\}_{j=1}^{l})$ changes into that of $(D^{\prime},\{S_j\}_{j=1}^{l+1})$ as follows. In this situation, $S_{l+1} \bigcap ({\overline{D}}^{{\mathbb{R}}^2}-D)$ is a three-point discrete set and we have $a_{1,1,S_{{l+1},(D,\{S_j\}_{j=1}^l)}}<a_{1,2,S_{{l+1},(D,\{S_j\}_{j=1}^l)}}<a_{1,3,S_{{l+1},(D

Figures (3)

  • Figure 1: Some cases of MBCC additions are presented. Black colored arcs represent ${\overline{D}}^{{\mathbb{R}}^2}-D$ or ${{\overline{D}}^{\prime}}^{{\mathbb{R}}^2}-D^{\prime}$. Gray colored regions represent part of $D$ or that of $D^{\prime}$. The Poincaré-Reeb graph of $(D,\{S_j\}_{j=1}^{l})$ and that of $(D^{\prime},\{S_j\}_{j=1}^{l+1})$ are also presented for each of the cases. We apply this rule for FIGUREs presented later.
  • Figure 2: An example for Case A-1 and an example for case B of SSCC additions are presented. The Poincaré-Reeb graph of $(D,\{S_j\}_{j=1}^{l})$ and that of $(D^{\prime},\{S_j\}_{j=1}^{l+1})$ are depicted for each case. Case A-2 is omitted.
  • Figure 3: The Poincaré-Reeb graph of $(D_{i+1},\{S_{j}\}_{j=1}^{i+1})$ and that of $(D_{0,i_1+2},\{S_{0,j}\}_{j=1}^{i_1+2})$ are depicted for an explicit case of an adjacent pair of an MBCC addition with an SSCC addition. Two additional vertices are added to edges existing already as subsets of the existing graph.

Theorems & Definitions (3)

  • Corollary 1
  • Corollary 2
  • proof