A comparability graph is a simple graph which admits a transitive orientation on its edges. Each one of such orientations defines a poset on the vertex set, and also it is said that this graph is the comparability graph of the poset. A treelike poset is a poset whose covering graph is a tree. Comparability graphs of arborescence posets are known as trivially perfect graphs. These have been char...

A comparability graph is a graph whose edges can be oriented transitively. Given a comparability graph G = (V, E) and an arbitrary edge ê ∈ E we explore the question whether the graph G − ê, obtained by removing the undirected edge ê, is a comparability graph as well. We define a new substructure of implication classes and present a complete mathematical characterization of all those edges.

We study the problem of adding edges to a given arbitrary graph so that the resulting graph is a comparability graph, called a comparability completion of the input graph. Computing a comparability completion with the minimum possible number of added edges is an NP-hard problem. Our purpose here is to add an inclusion minimal set of edges to obtain a minimal comparability completion, which mean...

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) ti...

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined inO(1) tim...

A graph is a P4-comparability graph if it admits an acyclic orientation of its edges which is transitive on every chordless path on four vertices. We give a characterization of P4-comparability graphs in terms of an auxiliary graph being bipartite. © 2006 Elsevier B.V. All rights reserved.