Cover-Incomparability Graphs of Posets

Abstract

Cover-incomparability graphs (C-I graphs, for short) are introduced, whose edge-set is the union of edge-sets of the incomparability and the cover graph of a poset. Posets whose C-I graphs are chordal (resp. distance-hereditary, Ptolemaic) are characterized in terms of forbidden isometric subposets, and a general approach for studying C-I graphs is proposed. Several open problems are also stated.

On the Complexity of Cover-Incomparability Graphs of Posets

“…It was already proved in [2] by a tedious case analysis. Together with our Theorem 1 it also easily follows from Gallai's characterization of comparability graphs by forbidden subgraphs [4].…”

“…[11,12], while the notion of coverincomparability graph is new. It was introduced in [2]. This notion was motivated by the theory of transit functions on posets.…”

“…This notion was motivated by the theory of transit functions on posets. It turns out that the underlying graph G TP of the standard transit function T P on the poset P is exactly the C-I graph of P (see [2] for details).…”

Incomparability Graphs of Lattices

“…Recently, Bostjan Bresar et.al., [2] introduced the cover incomparability graphs of posets and called these graphs as C − I graphs of P . They defined the graph in which the edge set is the union of the edge set of the corresponding covering graph and the corresponding incomparability graph.…”

Incomparability Graphs of Lattices II

“…Let R be a commutative ring with identity and let W (R) * be the set of all nonzero and nonunit elements of R. Two distinct vertices a and b in W (R) * are adjacent if and only if a / ∈ bR and b / ∈ aR. Recently, Bresar et al [3] introduced the cover incomparability graphs of posets and called these graphs as C − I graphs of P . They defined the graph in which the edge set is the union of the edge sets of the corresponding covering graph and the corresponding incomparability graph.…”