On superperfect noncomparability graphs

The class of superperfect graphs, which was previously studied by A. J. Hoffman, E. L. Johnson, and M. C. Golumbic, is a proper subclass of the class of perfect graphs; further, it properly contains the class of comparability graphs. In his book, Golumbic proves that, for split graphs, G is a comparability graph if and only if G is superperfect. Moreover, the fact that split graphs are exactly those graphs which are both triangulated and cotriangulated motivated Golumbic to ask if it is true or false that, for triangulated (or cotriangulated) graphs, G is a comparability graph if and only if G is superperfect. In the present paper, we determine those members of Gallai's list of minimal noncomparability graphs which are superperfect and, as a consequence, we find that the answer to the above question is “false” for triangulated and “true” for cotriangulated graphs.