Cover-Incomparability Graphs and 2-Colored Diagrams of Posets*

Abstract

As a continuation of the study of cover-incomparability graphs of posets (C-I graphs), the notion of 2-colored diagrams is introduced and used in characterizations of posets whose C-I graphs belong to certain natural classes of graphs. As a particular instance, posets whose C-I graphs are chordal are characterized using a single 2-colored diagram. Some other instances are characterized in a similar way.

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

“…2-coloured diagram P; in [12] we describe the family P by the Hasse diagram of initial poset P using normal edges, added by the bold edges between u i and v j (u i and v j are incomparable pairs) for all i and j. It follows that if there is a bold edge between an incomparable pair of elements u i and v j in P then either u i ⊲ v j or v j ⊲ u i ,which neither affect the covering nor the incomparability relation of any other pair of elements in P. Any subset of the set of bold edges can thus be chosen and removed arbitrarily to obtain one of the Hasse diagram of a poset from the family P. Hence one drawing, using normal and bold edges, suffices to describe all posets of P.…”

Characterizing Subclasses of Cover-Incomparability Graphs by Forbidden Subposets

“…Posets whose cover-incomparability graphs are chordal, Ptolemaic, distance-hereditary, claw-free or cographs were characterized in [1] and [4]. Unfortunately, there is a mistake that originated in [1] and continued in [4] and several statements from these papers do not hold as they are stated. In this paper we correct the mistake and reformulate the corresponding statements so that they hold.…”

“…Another approach is to study posets whose cover-incomparability graphs have certain property. Posets whose cover-incomparability graphs are chordal, Ptolemaic, distance-hereditary, claw-free or cographs were characterized in [1] and [4]. Unfortunately, there is a mistake that originated in [1] and continued in [4] and several statements from these papers do not hold as they are stated.…”

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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.

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