Graphs with independence number two

Characterize the graphs G with μ(G) = 2, beyond cycles, that are r-EKR.

Background

The survey notes that cycles provide an important class of graphs with independence number two and that several results establish the r-EKR property for cycles and related constructions.

It explicitly asks for other graphs with the same independence number that satisfy the property.

References

What other graphs $G$ (besides cycles) with $\alpha(G)=2$ are $r$-?

— A Survey of the Holroyd-Talbot Conjecture  (2501.16144 - Hurlbert, 27 Jan 2025) in Question 2(c), Section 5