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Discrete Moving Frames, Semi-Algebraic Invariants and the Graph Canonization Problem

Leonid Bedratyuk

TL;DR

This work reframes graph canonization as constructing a discrete moving frame for the finite group action of the pair group $S_n^{(2)}$ on edge weights, yielding a canonical representative via lexicographic orbit cross-sections. It moves beyond the traditional polynomial invariant approach, showing that the resulting invariants are semi-algebraic and form a complete set that separates isomorphism classes. By defining invariantization in this finite setting, the paper provides a rigorous theoretical bridge between invariant theory and practical canonization procedures, such as lexicographic labeling, and demonstrates how the canonical coordinates recover the graph's orbit uniquely. The results offer a conceptual framework that interprets canonization as orbit normalization and invariantization, with potential implications for analyzing the complexity and stability of canonical labeling algorithms.

Abstract

This paper develops an invariant--geometric interpretation of the canonization problem for simple undirected weighted graphs based on the {discrete moving frame method} for finite groups. We consider the action of the {pair group} $S_n^{(2)}$ on the space of edge weights of a graph. It is emphasized that the classical algebraic approach aimed at describing the ring of polynomial invariants of this action quickly becomes computationally impractical due to the explosive growth in the number and degrees of generators. The main result is a formalization of a canonical labeling of a graph as a {discrete moving frame} in the sense of Olver: a discrete orbit cross-section is fixed, in particular by a lexicographic rule, and for each configuration of edge weights one defines a permutation in $S_n^{(2)}$ that maps it to its canonical representative. The coordinates of the canonical representative are interpreted as a {complete system of invariants} for the action of $S_n^{(2)}$ that separates orbits, i.e., isomorphism classes of graphs. It is shown that the invariants obtained via such orbit canonization are of a non-algebraic nature and belong to the class of {semi-algebraic functions}. We do not propose a new computational algorithm; instead, we provide a rigorous theoretical foundation for the very concept of canonization by viewing it as a process of constructing a discrete moving frame and the corresponding system of semi-algebraic invariants.

Discrete Moving Frames, Semi-Algebraic Invariants and the Graph Canonization Problem

TL;DR

This work reframes graph canonization as constructing a discrete moving frame for the finite group action of the pair group on edge weights, yielding a canonical representative via lexicographic orbit cross-sections. It moves beyond the traditional polynomial invariant approach, showing that the resulting invariants are semi-algebraic and form a complete set that separates isomorphism classes. By defining invariantization in this finite setting, the paper provides a rigorous theoretical bridge between invariant theory and practical canonization procedures, such as lexicographic labeling, and demonstrates how the canonical coordinates recover the graph's orbit uniquely. The results offer a conceptual framework that interprets canonization as orbit normalization and invariantization, with potential implications for analyzing the complexity and stability of canonical labeling algorithms.

Abstract

This paper develops an invariant--geometric interpretation of the canonization problem for simple undirected weighted graphs based on the {discrete moving frame method} for finite groups. We consider the action of the {pair group} on the space of edge weights of a graph. It is emphasized that the classical algebraic approach aimed at describing the ring of polynomial invariants of this action quickly becomes computationally impractical due to the explosive growth in the number and degrees of generators. The main result is a formalization of a canonical labeling of a graph as a {discrete moving frame} in the sense of Olver: a discrete orbit cross-section is fixed, in particular by a lexicographic rule, and for each configuration of edge weights one defines a permutation in that maps it to its canonical representative. The coordinates of the canonical representative are interpreted as a {complete system of invariants} for the action of that separates orbits, i.e., isomorphism classes of graphs. It is shown that the invariants obtained via such orbit canonization are of a non-algebraic nature and belong to the class of {semi-algebraic functions}. We do not propose a new computational algorithm; instead, we provide a rigorous theoretical foundation for the very concept of canonization by viewing it as a process of constructing a discrete moving frame and the corresponding system of semi-algebraic invariants.
Paper Structure (14 sections, 2 theorems, 88 equations)

This paper contains 14 sections, 2 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The Classical Algebraic Approach to Polynomial Graph Invariants
  3. The group $S_n^{(2)}$ and its action on a graph
  4. Polynomial invariants of the group $S_n^{(2)}$
  5. The Moving Frame Method for Finite Groups
  6. Discrete moving frames
  7. Completeness of the invariantized coordinates
  8. Illustration of the method for the symmetric group $S_n$
  9. A complete system of semi-algebraic invariants for $S_n^{(2)}$.
  10. Orbit canonization
  11. Invariantization and a complete system of invariants.
  12. Semi-algebraic nature of the obtained invariants
  13. Example for $n=4$
  14. Conclusions

Key Result

Theorem 3.1

Let a finite group $G$ act on a manifold $\mathcal{M}$, and let $\mathcal{K}\subset\mathcal{M}$ be a cross-section, while $\mathcal{M}_{\mathrm{reg}}\subset\mathcal{M}$ is a subset on which each $G$-orbit intersects $\mathcal{K}$ in exactly one point. Let $\rho:\mathcal{M}_{\mathrm{reg}}\to G$ be a Consider the map Then:

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof