Characterizing subclasses of cover-incomparability graphs by forbidden subposets

TL;DR

This paper refines the understanding of cover-incomparability graphs of finite posets by analyzing forbidden subposet characterizations and correcting previous inaccuracies through detailed examples and proofs.

Contribution

It provides revised, valid forbidden subposet characterizations of subclasses of cover-incomparability graphs, clarifying previous misconceptions.

Findings

Identifies issues with previous forbidden subposet characterizations

Provides corrected characterizations with proofs

Highlights the importance of isometric subsets in these characterizations

Abstract

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.