Optimal graph joining with applications to isomorphism detection and identification

TL;DR

This work develops an optimal transport–based framework for comparing undirected weighted graphs with vertex labels via graph joinings. The core OGJ problem minimizes the cost ⟨c, r_γ⟩ over the convex set of weight joinings 𝒥(α, β), producing a linear programming formulation whose solutions correspond to transport plans between graphs. The authors establish strong links to graph isomorphism, providing sufficient conditions under which OGJ detects and identifies isomorphism for various graph families (including trees and forests) using informative labeling schemes and process-level labels. They introduce an extension principle (magic decompositions) to propagate detection/identification through graph constructions and relate OGJ to classical GI tools (WL, convex relaxations), as well as to the WL/NetOTC hierarchy. Overall, the framework offers a flexible, mathematically principled approach to GI that integrates local vertex information with global coupling structure and yields tractable, stable computations.

Abstract

We introduce an optimal transport based approach for comparing undirected graphs with non-negative edge weights and general vertex labels, and we study connections between the resulting linear program and the graph isomorphism problem. Our approach is based on the notion of a joining of two graphs $G$ and $H$, which is a product graph that preserves their marginal structure. Given $G$ and $H$ and a vertex-based cost function $c$, the optimal graph joining (OGJ) problem finds a joining of $G$ and $H$ minimizing degree weighted cost. The OGJ problem can be written as a linear program with a convex polyhedral solution set. We establish several basic properties of the OGJ problem, and present theoretical results connecting the OGJ problem to the graph isomorphism problem. In particular, we examine a variety of conditions on graph families that are sufficient to ensure that for every pair of graphs $G$ and $H$ in the family (i) $G$ and $H$ are isomorphic if and only if their optimal joining cost is zero, and (ii) if $G$ and $H$ are isomorphic, the the extreme points of the solution set of the OGJ problem are deterministic joinings corresponding to the isomorphisms from $G$ to $H$.

Figures (6)

• Figure 1: (a) A labeled graph with vertex labels in $\{0,1,2\}$ and edge weights in $\{\tfrac{1}{3}, \tfrac{1}{12}\}$, and (b) an unlabeled graph with edge weights in $\{\tfrac{1}{4}, \tfrac{1}{8}\}$. Note that the edge weights drawn here are equal to the weight function value $\alpha(u,u')$ for each ordered pair $(u,u')$ corresponding to an edge.
• Figure 2: Illustrations of three graph joinings, with the vertex sets of $G$ and $H$ drawn on the horizontal and vertical axes, respectively, and a graph joining drawn on the grid representing the product of their vertex sets. Edge weights and vertex labels are not drawn. Note that (b) and (c) depict two distinct graph joinings of the same pair of marginal graphs.
• Figure 3: A labeled graph with primary labels $1,2,3,4$. The discrete degree label function takes values $1, 2, 2, 1$, and the multiweight label function takes values: $\{ \! \{ \tfrac{1}{3} \} \! \}$, $\{ \! \{ \tfrac{1}{3},\tfrac{1}{12}\} \! \}$, $\{ \! \{ \tfrac{1}{12}, \tfrac{1}{12} \} \! \}$, and $\{ \! \{ \tfrac{1}{12} \} \! \}$.
• Figure 4: A graph with vertices labeled by the discrete degree label function. For this example the discrete degree label function is not injective, while direct calculation shows that the process-level label function is injective.
• Figure 5: An illustration of a gluing. (a) Two graphs $G_1$ and $G_2$, and the set $M$, which contains only a single element (pictured as a square). Here $f_1$ and $f_2$ map all the leaves of the two graphs to the unique element of $M$. (b) The gluing of $G_1$ and $G_2$ along $M$.

Theorems & Definitions (118)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.6
• Definition 3.1
• Example 3.2