When Partial Atom Maps Can Be Uniquely Completed: New Algorithmic Guarantees

Chemical reaction databases usually record the structures of reactants and products and their stoichiometry, but they rarely specify which atom in a reactant becomes which atom in a product. These atom-to-atom maps, or AAMs, are essential for applications such as chemical synthesis planning in organic chemistry and the analysis of isotope labeling experiments in modern metabolomics. Because the maps are generally missing from the databases, they must be reconstructed computationally. A new study published in Algorithms for Molecular Biology by Marcos E. González Laffitte, Tieu-Long Phan and Peter F. Stadler addresses a precise mathematical question at the heart of this reconstruction problem: when can a partially known atom map be extended to a complete one, and in what sense is that extension unique?

The authors note that chemically correct AAMs are ultimately determined by quantum-mechanical phenomena, and therefore cannot be reliably computed by solving graph-theoretical optimization problems defined only by the reactant and product structures. Their proposed way around this difficulty is to shift the focus: first identify a partial AAM that contains the reaction center, meaning the atoms incident with all bonds that change during the reaction, and then extend that partial map to the full reaction. The complete AAM of a reaction is faithfully represented by the Imaginary Transition State (ITS) graph, which provides a convenient graph-theoretic framework for analyzing when and how such an extension is possible.

Within this framework the researchers establish a clean characterization. They show that a unique extension of a partial AAM exists whenever, and only if, the partial maps cover the reaction center. In other words, coverage of the reaction center is both necessary and sufficient for uniqueness, a result that turns an intuitive chemical notion into a provable graph-theoretic condition. The uniqueness results are further generalized to partial AAMs in situations where hydrogen atoms are not represented explicitly, a common simplification in practical chemical databases that complicates the mapping problem.

For the computational side, the study shows that when hydrogens are implicit, the extension of a partial map can be computed by solving a constrained graph-isomorphism search between specific subgraphs of ITS graphs. This reformulation connects the reconstruction task to a well-studied algorithmic problem, giving a concrete route from partial information to a complete reaction map. The authors close by benchmarking different tools for this task, providing a practical comparison of available methods. The work thus offers both a theoretical guarantee — uniqueness tied to reaction-center coverage — and an algorithmic strategy for computing extensions, situating the results within the broader effort to reconstruct atom mappings computationally for synthesis planning and isotope-labeling studies.

Subject of Research: Uniqueness conditions and algorithms for extending partial atom-to-atom maps of chemical reactions

Article Title: Extension of partial atom-to-atom maps: uniqueness and algorithms

Article References: Laffitte, M. E. G., Phan, T.-L., & Stadler, P. F. (2026). Extension of partial atom-to-atom maps: uniqueness and algorithms. Algorithms for Molecular Biology, 21(1), Article 10. https://doi.org/10.1186/s13015-026-00300-5

Image Credits: AI Generated

DOI: 10.1186/s13015-026-00300-5

Keywords: atom-to-atom mapping, chemical reactions, Imaginary Transition State graph, graph isomorphism, reaction center, chemical databases, metabolomics, synthesis planning, algorithms, computational chemistry, Extension, partial