Can transitive orientation make sandwich problems easier?

A graph Gs=(V,Es)Gs=(V,Es) is a sandwich for a pair of graphs Gt=(V,Et)Gt=(V,Et) and G=(V,E)G=(V,E) if Et⊆Es⊆EEt⊆Es⊆E. A sandwich problem asks for the existence of a sandwich graph having an expected property. In a seminal paper, Golumbic et al. [Graph sandwich problems, J. Algorithms 19 (1995) 449–473] present many results on sub-families of perfect graphs. We are especially interested in comparability (resp., co-comparability) graphs because these graphs (resp., their complements) admit one or more transitive orientations (each orientation is a partially ordered set or poset). Thus, fixing the orientations of the edges of GtGt and G restricts the number of possible sandwiches. We study whether adding an orientation can decrease the complexity of the problem. Two different types of problems should be considered depending on the transitivity of the orientation: the poset sandwich problems and the directed sandwich problems. The orientations added to both graphs G   and GsGs are transitive in the first type of problem but arbitrary for the second type.