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Realizations of planar graphs as Poincar'e-Reeb graphs of refined algebraic domains

Naoki Kitazawa

Abstract

Algebraic domains are regions in the plane surrounded by mutually disjoint non-singular real algebraic curves. Poincar'e-Reeb Graphs of them are graphs they naturally collapse: such graphs are formally formulated by Sorea, for example, around 2020. Their studies found that nicely embedded planar graphs are Poincar'e-Reeb graphs of some algebraic domains. These graphs are generic with respect to the projection to the horizontal axis. Problems, methods and results are elementary and natural and they apply natural approximations nicely for example. We present our new approach to extension of the result to a non-generic case and an answer. We first formulate generalized algebraic domains, surrounded by non-singular real algebraic curves which may intersect with normal crossings. Such domains and certain classes of them appear in related studies of graphs and regions surrounded by algebraic curves explicitly.

Realizations of planar graphs as Poincar'e-Reeb graphs of refined algebraic domains

Abstract

Algebraic domains are regions in the plane surrounded by mutually disjoint non-singular real algebraic curves. Poincar'e-Reeb Graphs of them are graphs they naturally collapse: such graphs are formally formulated by Sorea, for example, around 2020. Their studies found that nicely embedded planar graphs are Poincar'e-Reeb graphs of some algebraic domains. These graphs are generic with respect to the projection to the horizontal axis. Problems, methods and results are elementary and natural and they apply natural approximations nicely for example. We present our new approach to extension of the result to a non-generic case and an answer. We first formulate generalized algebraic domains, surrounded by non-singular real algebraic curves which may intersect with normal crossings. Such domains and certain classes of them appear in related studies of graphs and regions surrounded by algebraic curves explicitly.

Paper Structure

This paper contains 8 sections, 1 theorem, 3 figures.

Table of Contents

  1. Introduction.
  2. Our notation on topological spaces, manifolds and graphs.
  3. Refined algebraic domains.
  4. Our main result.
  5. Organization of our paper and our main work.
  6. A proof of Theorem \ref{['thm:1']}.
  7. Relations with Reeb graphs of real algebraic functions.
  8. Conflict of interest and Data availability.

Key Result

Theorem 1

For any finite and connected graph $G$ and a piecewise smooth function $c_G:G \rightarrow \mathbb{R}$ such that the restriction ${c_G} {\mid}_{e}$ is injective for each edge $e$ of $G$, we can canonically give $G$ the structure of a V-digraph by the function $c_G$. We also assume the following. Then we have a refined algebraic domain $D_{G}$ and its Poincaré-Reeb V-digraph of $D_{G}$ and the V-di

Figures (3)

  • Figure 1: Around a vertex $v_p \in e_G(G)$ and $N_v$. The set $N_v \bigcap (e_G(G) \bigcup S_p)$ (, in the upper part,) is changed.
  • Figure 2: Around a connected component of $N_v-G_{\epsilon}$ whose closure contains no point of the form $(x,p_v \pm {\epsilon}_2)$. The blue region shows (the interior of) $M_D$ partially and the red ellipsoid is added.
  • Figure 3: Around a vertex of degree $1$ of $e_G(G)$. The blue region shows (the interior of) $M_D$, containing the graph $e_G(G)$, partially. The red circle (with the disk bounded by this) is added.

Theorems & Definitions (2)

  • Theorem 1
  • proof : A proof of Theorem \ref{['thm:1']}