Other representations with one-label doubling behavior

Determine whether transitive permutation-group representations beyond regular and Frobenius representations, or all transitive representations of additional group families, exhibit the property that removing a single derangement label doubles the independence number and supports analogous robustness results.

Background

The paper studies how deleting inverse-closed sets of derangements from the connection set of a permutation group's derangement graph changes its independence number. For regular and Frobenius representations, deleting one label can double the independence number, and related behavior is established for several generalized dihedral and dicyclic representations.

The authors ask whether this phenomenon extends to other classes of permutation representations or group families, thereby identifying broader structural mechanisms behind the observed jump in independence number.

References

Question 7.1. Are there other transitive group representations that have similar properties-either with respect to a class of representations (as for regular and Frobenius representations) or with respect to all transitive representations of some family of groups (like the generalised dihedral groups over abelian groups that include odd permutations)?

— A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups  (2502.14582 - Gunderson et al., 20 Feb 2025) in Question 7.1, Section 7