  In this paper we continue investigations of cover-incomparability graphs of
finite partially ordered sets (see \cite{Bres,Bres2,Bres3,Bres4} and
\cite{Max,MaxDH}). We consider in some detail the distinction between
cover-preserving subsets and isometric subsets of a partially ordered set. This
is critical to understanding why forbidden subposet characterizations of
certain classes of cover-incomparability graphs in \cite{Bres} and \cite{Bres3}
are not valid as presented. Here we provide examples, investigate the root of
the difficulties, and formulate and prove valid revisions of these
characterizations.

Combinatorics;Discrete Mathematics