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arXiv · 2511.14862

Optimal graph joining with applications to isomorphism detection and identification

Phuong N. Hoàng·Kevin McGoff·Andrew B. Nobel·Yang Xiang·Bongsoo Yi
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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 $H$.

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Phuong N. Hoàng, Kevin McGoff, Andrew B. Nobel, Yang Xiang, Bongsoo Yi. 2025-11-18. Optimal graph joining with applications to isomorphism detection and identification. https://arxiv.org/abs/2511.14862

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