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Canonical decompositions and algorithmic recognition of spatial graphs

Stefan Friedl, Lars Munser, José Pedro Quintanilha, Yuri Santos Rego

Abstract

We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colorings, edge colorings and/or edge orientations. We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-separable and have no cut vertices, in a suitable topological sense. Then we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

Canonical decompositions and algorithmic recognition of spatial graphs

Abstract

We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colorings, edge colorings and/or edge orientations. We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-separable and have no cut vertices, in a suitable topological sense. Then we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

Paper Structure

This paper contains 19 sections, 54 theorems, 20 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Our main result
  3. Matveev's Recognition Theorem
  4. Decomposition results
  5. The piecewise-linear setting
  6. Structure of the paper
  7. Basic definitions
  8. The disjoint union of spatial graphs
  9. Assembling spatial graphs through disjoint unions
  10. Decomposing spatial graphs as disjoint unions
  11. The vertex sum of spatial graphs
  12. Defining the vertex sum
  13. Iterated vertex sums and trees of spatial graphs
  14. Decomposing pieces as trees of blocks
  15. Extension to decorated graphs
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exists an algorithm for determining whether two spatial graphs are isomorphic.

Figures (16)

  • Figure 1.1: Two pairs of (decorated) spatial graphs with the same combinatorial structure, but a priori different topology.
  • Figure 3.1: The proof of associativity of the disjoint union, with ambient spheres and enclosing balls depicted one dimension below. Top: the ambient spheres and enclosing balls for the spatial graphs $\Gamma_i$. Bottom: the spatial graph $\Gamma_1 \sqcup_{f_1} \Gamma_2 \sqcup_{f_3} \Gamma_3$ in its ambient sphere $B_1 \cup_{f_1} (B_{21} \cap B_{23}) \cup_{f_3} B_3$.
  • Figure 3.2: The notion of non-separability of spatial graphs does not coincide with the notion of connectedness. The depicted spatial graph $\Gamma$ is non-separable, but its support $|\Gamma|$ and underlying graph $\langle \Gamma\rangle$ are disconnected.
  • Figure 4.1: The setup of Lemma \ref{['lem.annulus']}. The PL annulus $A$ "interpolates" between the PL circles $\gamma, \delta$.
  • Figure 4.2: The $3$-balls $\overline B$ and $\overline{B'}$ in the setup of the proof of Lemma \ref{['prop.penclosingball']}. We emphasize the action of $\Phi_{\overline B}$ on $vD$ as the cone of a PL homeomorphism $D \to D'$.
  • ...and 11 more figures

Theorems & Definitions (118)

  • Theorem 1.1: Algorithmic detection of spatial graphs
  • Theorem 1.2: Canonical decomposition as a tree of blocks
  • Theorem 1.3: Uniqueness of spatial trees
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1: Disc Theorem RourkeSanderson
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4: Uniqueness of enclosing balls
  • ...and 108 more