Whether kernel-M-solvability characterizes perfect graphs

Determine whether kernel-$M$-solvability characterizes perfect graphs; equivalently, establish whether every kernel-$M$-solvable graph is perfect.

Background

Kernel-solvability characterizes perfect graphs by the Boros–Gurvich theorem. Kernel-MM-solvability is a related, more restrictive orientation property, and several special cases of the perfect-graph question were historically studied through it. The paper explicitly leaves unresolved whether kernel-MM-solvability already gives the same characterization of perfectness.

References

More importantly, we know by the Boros--Gurvich theorem that a graph is perfect if and only if it is kernel-solvable; it is still possible that kernel-$M$-solvability already characterizes perfectness.

— Revisiting classical results on kernels in digraphs  (2502.02482 - Langlois et al., 4 Feb 2025) in Section 4, first paragraph