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arXiv · 2610.14135

Reconstruction and Symmetry of Two-Token Graphs

Yuanming Luo·Hao Hu
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Abstract

We present a proof of the two-token graph reconstruction conjecture. Every finite simple graph $G,H$ of order at least three, $F_2(G) \cong F_2(H)$ if and only if $G \cong H$. We also classify the automorphism groups of two-token graphs whose connected roots have order at least five. Combining these results, we show that, for connected graphs $G,H$ of order at least five, every isomorphism $ϕ:F_2(G)\to F_2(H)$ admits a factorization $ϕ=η\circσ^{(2)}$, where $σ^{(2)}$ is induced by a graph isomorphism $σ:G\to H$, and $η\in \operatorname{Aut}(F_2(H))$ is a product of twin-cut switches or Cartesian exchanges.

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Yuanming Luo, Hao Hu. 2026-10-09. Reconstruction and Symmetry of Two-Token Graphs. https://arxiv.org/abs/2610.14135

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