On Dually-CPT and Strongly-CPT Posets

poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in a tree and two elements are comparable in the poset if the corresponding paths are related by the inclusion relation. Recently Alcón, Gudiño and Gutierrez (Discrete Applied Math. 245 , 139–147, 2018 ) introduced proper subclasses of CPT posets, namely dually-CPT, and strongly-CPT (or strong-CPT). A poset P is dually-CPT, if P and its dual P^d both admit a CPT-representation. A poset P is strongly-CPT, if P and all the posets that share the same underlying comparability graph admit a CPT-representation. Where as the inclusion between dually-CPT and CPT was known to be strict. It was raised as an open question by Alcón, Gudiño and Gutierrez (Discrete Applied Math. 245 , 139–147, 2018 ) whether strongly-CPT was a strict subclass of dually-CPT. We provide a proof that both classes actually coincide.