A corrected version of the Duchet kernel conjecture

In 1980, Piere Duchet conjectured that odd-directed cycles are the only edge minimal kernel-less connected digraphs, i.e. in which after the removal of any edge a kernel appears. Although this conjecture was disproved recently by Apartsin et al. (1996), the following modification of Duchetʹs conjecture still holds: odd holes (i.e. odd-non-directed chordless cycles of length 5 or more) are the only connected graphs which are not kernel-solvable but after the removal of any edge the resulting graph is kernel-solvable.