Cocomparability of the next power of an AT-free power

Determine whether, for every graph G and every integer k, the cocomparability of G^k being AT-free implies that G^{k+1} is cocomparability.

Background

The paper studies the power-AT and power-cocomparability indices of a graph G, defined as the smallest exponents k for which Gk is AT-free or cocomparability, respectively. Although powers of AT-free graphs are known to be cocomparability for exponent greater than one, the authors identify an unresolved question concerning the immediately subsequent power of a graph whose kth power is AT-free. The paper proves a weaker but sufficient bound: if Gk is AT-free, then a higher power, specifically G{2k} in the relevant path-length setting, is cocomparability.

References

We do not know if $G{k+1}$ is a cocomparability graph for a graph $G$ whose $k{th}$ power $Gk$ is an AT-free graph.

— Graph parameters that are coarsely equivalent to path-length  (2503.05661 - Dragan et al., 7 Mar 2025) in Section 3.1, subsection “Powers of AT-free graphs, k-dominating pairs and k-dominating shortest paths”