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

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.