Characterization of distinguishing switching methods

Determine which $(Q,\Gamma)$-switching methods are distinguishing, where a switching method is distinguishing when the group $Aut_Q(\Gamma)$ acts faithfully on the set of $Q$-respecting vectors.

Background

The paper introduces distinguishing as a technical condition needed to obtain the precise asymptotic enumeration in Theorem 1. The condition prevents nontrivial automorphisms of the switching graph from fixing all admissible attachment vectors, which would otherwise cause complications in counting isomorphism classes.

The authors verify the condition for several families of switching methods later in the paper, but do not provide a general characterization for arbitrary pairs $(Q,\Gamma)$. Consequently, the theorem retains distinguishing as an assumption.