On dually-CPT and strong-CPT posets

Abstract

A 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 and only if the corresponding paths are related by the inclusion relation. Recently Alcón, Gudiño and Gutierrez introduced proper subclasses of CPT posets, namely dually-CPT, and strongly-CPT. A poset $\mathbf{P}$ is dually-CPT, if and only if $\mathbf{P}$ and its dual $\mathbf{P}^{d}$ both admit a CPT representation. A poset $\mathbf{P}$ is strongly-CPT, if and only if $\mathbf{P}$ 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 whether strongly-CPT was a strict subclass of dually-CPT. We provide a proof that both classes actually coincide.