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Two-fold automorphisms (or “TF-isomorphisms”) of graphs are a generalisation of automorphisms. Suppose \(\alpha, \beta\) are two permutations of \(V = V(G)\) such that for any pair \((u,v)\), \(u, v \in V\), \((u,v)\) is an arc of \(G\) if and only if \((\alpha(u), \beta(v))\) is an arc of \(G\). Such a pair of permutations is called a two-fold automorphism of \(G\). These pairs form a group that is called the two-fold automorphism group. Clearly, it contains all the pairs \((\alpha, \alpha)\) where \(\alpha\) is an automorphism of \(G\). The two-fold automorphism group of \(G\) can be larger than \(\text{Aut}(G)\) since it may contain pairs \((\alpha, \beta)\) with \(\alpha \neq \beta\). It is known that when this happens, \(\text{Aut}(G) \times \mathbb{Z}_2\) is strictly contained in \(\text{Aut}(G \times K_2)\). In the literature, when this inclusion is strict, the graph \(G\) is called unstable.
Now let \(\Gamma \leq S_V \times S_V\). A two-fold orbital (or “TF-orbital”) of \(F\) is an orbit of the action \((\alpha, \beta) : (u,v) \mapsto (\alpha(u), \beta(v))\) for \((\alpha, \beta) \in \Gamma\) and \(u,v \in V\). Clearly, \(\Gamma\) is a subgroup of the TF-automorphism group of any of its TF-orbitals. We give a short proof of a characterization of TF-orbitals which are disconnected graphs and prove that a similar characterization of TF-orbitals which are digraphs might not be possible. We shall also show that the TF-rank of \(\Gamma\), that is the number of its TF-orbitals, can be equal to \(1\) and we shall obtain necessary and sufficient conditions on I for this to happen.