On dually-CPT and strong-CPT posets

TL;DR

This paper investigates specialized classes of containment of paths in a tree (CPT) posets, proving that the classes of dually-CPT and strongly-CPT are actually the same, resolving an open question in the field.

Contribution

The paper proves that the classes of dually-CPT and strongly-CPT posets are equivalent, clarifying the relationship between these subclasses of CPT posets.

Findings

Dually-CPT and strongly-CPT classes coincide.

Resolved an open question about the strictness of subclass inclusion.

Enhanced understanding of CPT poset subclasses.

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 Alc\'on, Gudi\~{n}o 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 Alc\'on, Gudi\~{n}o and Gutierrez whether strongly-CPT was a strict subclass of dually-CPT. We provide a proof that both classes actually coincide.