Characterizing Subclasses of Cover-Incomparability Graphs by Forbidden Subposets

Abstract

In this paper we continue investigations of cover-incomparability graphs of finite partially ordered sets (see [1,2,3,4] and [6,7]). 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 [1] and [3] are not valid as presented.Here we provide examples, investigate the root of the difficulties, and formulate and prove valid revisions of these characterizations.

Composition and product of cover-incomparability graphs

“…The ◁-preserving subposet was introduced in [5] for proving the Theorem 2.4 in [8]. We quote the theorem from [5].…”

2-Colored and 3-Colored Diagrams of Posets in Cover-incomparability Graphs

“…Theorem 1: (Theorem 1, [7]): Let G be a class of graphs with a forbidden induced subgraphs characterization. Let P = {P | P is a poset with G }.…”

Forbidden 3-Colored Posets of Cover-Incomparable Line Graphs

“…In a similar way the dual of a 3-colored diagram is also meaningful and represents a collection of ⊲-preserving dual subposets. Theorem 2: (Theorem 1, [8]): Let G be a class of graphs with a forbidden induced subgraphs characterization. Let P = {P | P is a poset with G }.Then P has a characterization by forbidden ⊲ -preserving subposets.…”