Reconstruction and Symmetry of Two-Token Graphs

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 $\phi :F_2(G)\to F_2(H)$ admits a factorization $\phi =\eta \circ {\sigma }^{(2)}$, where ${\sigma }^{(2)}$ is induced by a graph isomorphism $\sigma :G\to H$, and $\eta \in \mathrm{Aut}(F_2(H))$ is a product of twin-cut switches or Cartesian exchanges.