Determine whether the next graph power is cocomparability

Determine whether, for every graph G whose kth power G^k is AT-free, the (k+1)st power G^{k+1} is necessarily a cocomparability graph.

Background

The paper uses graph powers to relate path-length to AT-free and cocomparability graph structure. It recalls that AT-free graph powers have strong structural properties, but the authors do not establish the corresponding one-step implication for cocomparability graphs.

The unresolved question asks whether AT-freeness of G^k guarantees cocomparability of the immediately higher power G^{k+1}. The paper instead proves the weaker quantitative result that G^{2λ} is cocomparability when the path-length of G is λ.