**Containment Graphs of Paths in a Tree**

**Liliana Alcon**

** **

**Abstract**

A *containment model* of a poset (X, <) maps each element *x* of *X* into a set *M _{x}* in such a way that

*x*<

*y*if and only if

*M*is a proper subset of

_{x}*M*. It is well known that posets admitting a containment model mapping vertices into intervals of the line (

_{y}*CI*posets for short) are the posets with dimension at most 2; thus, if a transitive orientation of a comparability graph

*G*is a

*CI*poset then any other transitive orientation of

*G*is also a

*CI*poset. Comparability graphs of

*CI*posets were shown to be the permutation graphs.

Generalizing this ideas, M. Golumbic, first in a work with E. Scheinerman and then in the book with A. Trenk, suggested the study of posets admitting a containment model mapping vertices into paths of a tree and their comparability graphs, called *CPT* posets and *CPT* graphs, respectively.

In this talk, based on a joint work with N. Gudiño and M. Gutierrez, I will present our first results on this line. Answering a question posed by J. Spinrad, we proved that, unlike what happens with *CI* posets, the dimension and the interval dimension of *CPT* posets is unbounded. On the other hand, we will see that the dimension of a *CPT* poset is at most the number of leaves of the tree used in the containment model. An example of a *CPT* poset *P* whose dual *P ^{d}* is non

*CPT*motivated us to introduce the notion of

*dually-CPT*and

*strong-CPT*. I will show that trees are

*strong-CPT*and will give a characterization of

*CPT*(also

*dually-CPT*and

*strong-CPT*) split posets by a finite family of forbidden subposets. Also we will consider the problem of characterizing

*CPT*posets restricted to posets whose comparability graphs are

*k*-trees. Several open problems will be posed.