[Paper Review] An Efficient Algorithm to Recognize Locally Equivalent Graphs in Non-Binary Case

Mohsen Bahramgiri, Salman Beigi|ArXiv.org|Feb 9, 2007

Graph Labeling and Dimension Problems5 references3 citations

TL;DR

This paper presents the first efficient polynomial-time algorithm to determine local equivalence between labeled graphs over finite fields of odd order, generalizing binary local complementation. By leveraging isotropic systems and algebraic structure of solution spaces, the method reduces the problem to checking affine subspaces of bounded codimension, enabling an O(n⁵) solution via basis enumeration and constraint verification.

ABSTRACT

Let $v$ be a vertex of a graph $G$. By the local complementation of $G$ at $v$ we mean to complement the subgraph induced by the neighbors of $v$. This operator can be generalized as follows. Assume that, each edge of $G$ has a label in the finite field $\mathbf{F}_q$. Let $(g_{ij})$ be set of labels ($g_{ij}$ is the label of edge $ij$). We define two types of operators. For the first one, let $v$ be a vertex of $G$ and $a\in \mathbf{F}_q$, and obtain the graph with labels $g'_{ij}=g_{ij}+ag_{vi}g_{vj}$. For the second, if $0 eq b\in \mathbf{F}_q$ the resulted graph is a graph with labels $g''_{vi}=bg_{vi}$ and $g''_{ij}=g_{ij}$, for $i,j$ unequal to $v$. It is clear that if the field is binary, the operators are just local complementations that we described. The problem of whether two graphs are equivalent under local complementations has been studied, \cite{bouchalg}. Here we consider the general case and assuming that $q$ is odd, present the first known efficient algorithm to verify whether two graphs are locally equivalent or not.

Motivation & Objective

• To develop an efficient algorithm for determining local equivalence of labeled graphs over finite fields of odd order, extending beyond the binary case.
• To address the challenge that non-binary local equivalence involves non-linear systems of equations, unlike the linearizable binary case.
• To establish a geometric framework using isotropic systems to model graph equivalence and exploit structural properties of solution spaces.
• To prove that if two graphs are locally equivalent, their solution space contains a large affine subspace of bounded codimension (≤5), enabling efficient search.

Proposed method

• Introduce isotropic systems as geometric objects to model local equivalence, defining equivalence via solvability of a system of algebraic equations.
• Associate each labeled graph with an isotropic system, showing that graph local equivalence corresponds to isotropic system equivalence.
• Formulate the local equivalence problem as solving a system of linear and quadratic equations over F_q, with variables representing transformation parameters.
• Prove that solution spaces for equivalent graphs contain affine subspaces of codimension at most 5, enabling efficient search.
• Use a basis of the solution space of linear equations (5) and test all linear combinations of at most five basis vectors to satisfy the quadratic constraint (6).
• Apply pivoting methods to compute the basis in O(n⁴) time and verify the quadratic condition in O(n⁵) time, yielding an overall O(n⁵) algorithm.

Experimental results

Research questions

• RQ1 Can local equivalence of non-binary labeled graphs over odd finite fields be decided efficiently, given the non-linear nature of the associated equations?
• RQ2 What structural properties do solution spaces of the equivalence equations exhibit when two graphs are locally equivalent?
• RQ3 How can the solution space of the equivalence system be exploited to reduce the search space for valid transformations?
• RQ4 Is there a uniform bound on the codimension of the solution space that enables efficient enumeration of candidate transformations?
• RQ5 Can the problem be reduced to checking a small number of linear combinations of basis vectors to verify equivalence?

Key findings

• The algorithm runs in O(n⁵) time, providing the first known polynomial-time solution for local equivalence in the non-binary case.
• If two graphs are locally equivalent, their associated solution space contains an affine subspace of codimension at most 5 within the full solution space.
• The solution space structure ensures that a solution can always be found as a linear combination of at most five basis vectors from a computed basis.
• The method reduces the search space for valid transformations by exploiting geometric properties of isotropic systems and bounded codimension.
• The algorithm is efficient and generalizes prior binary results by handling the more complex non-linear structure of odd-characteristic fields.
• The framework establishes a rigorous connection between graph local equivalence and the geometry of isotropic systems over finite fields.