r-EKR property for well-covered graphs

Determine whether every well-covered graph is r-EKR for every admissible r.

Background

The survey explains that well-covered graphs are natural candidates because every maximal independent set has the same size, matching the extremal parameter appearing in the Holroyd–Talbot conjecture.

Several well-covered graph classes are known to satisfy related EKR properties, but the general class remains unresolved; the rook graph K_m □ K_m is highlighted as a particularly attractive test case.