Perfectly Orderable P4 Composition

A P4 composition scheme is as follows: start with two graphs with vertex sets of different colours, say black • and white ○ select a set of allowable four-vertex colour sequences, for example {••••, ○○○○, •○○•, ○••○}; add edges between the graphs so that in the composed graph each four-vertex induced path is coloured with an allowable sequence. Answering a question of Chvétal, we determine all such schemes which preserve perfect orderability.