Cographs Which are Cover-Incomparability Graphs of Posets

Characterizing Subclasses of Cover-Incomparability Graphs by Forbidden Subposets

“…7. It is easy to see that it is a counterexample to Theorem 4.1 [3]. Indeed, P contains neither any of the posets Q 1 , Q 2 , .…”

“…Similarly, it is not possible that x ≺ u, v ≺ x or w ≺ x. Thus there are only five cases to distinguish: Now let us restate the corresponding statements from [1] and [3]. We skip their proofs as they are the same as the ones presented in [1] and [3], the only mistake was claiming that the forbidden subposets must be isometric subposets of P .…”

“…In Section 2 we give an overview of terminology and basic properties of cover-incomparability graphs. In Section 3 we present counterexamples to Theorem 4.1 from [3], Lemma 4.4 and 4.5 from [1] and to Proposition 5.1 from [1]. In Section 4 we show that the mistake originated in Theorem 2.4 of [1].…”

“…In [6] it was proved that the recognition problem for cover-incomparability graphs is in general NP-complete. On the other hand there are classes of graphs (such as trees, Ptolemaic graphs, distance-hereditary graphs, block graphs, split graphs or k-trees) for which the recognition problem can be solved in linear time (see [2,3,7,8] for details and proofs).…”

“…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.…”