Dushnik and Miller defined the dimension of a partial order P as the minimum number of linear orders whose intersection is P. Ken Bogart asked if the dimension of a partial order is an invariant of the associated comparability graph. In this paper we answer Bogart's question in the affirmative. The proof involves a characterization of the class of comparability graphs defined by Aigner and Prin...

Lexicographic depth first search (LexDFS) is a graph search protocol which has already proved to be a powerful tool on cocomparability graphs, a subfamily of perfect graphs. Cocomparability graphs have been well studied through their complements (comparability graphs) and their corresponding posets. Recently however LexDFS has led to a number of elegant near linear time algorithms on cocomparab...

Given a class of graphs G, a graphG is a probe graph of G if its vertices can be partitioned into a set of probes and an independent set of nonprobes such that G can be embedded into a graph of G by adding edges between certain nonprobes. If the partition of the vertices is part of the input, we call G a partitioned probe graph of G. In this paper we show that there exists a polynomial-time alg...

We consider the problem of recognizing whether a simple undirected graph is a P4-comparability graph. This problem has been considered by Hoàng and Reed who described an O(n)-time algorithm for its solution, where n is the number of vertices of the given graph. Faster algorithms have recently been presented by Raschle and Simon and by Nikolopoulos and Palios; the time complexity of both algorit...

We consider two problems pertaining to P4-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P4-comparability graph and the problem of producing an acyclic P4-transitive orientation of a P4-comparability graph. These problems have been considered by Hoàng and Reed who described O(n) and O(n)-time algorithms for their solution respectively, where n is...

It is well known that the comparability graph of any partially ordered set of n elements contains either a clique or an independent set of size at least pn. In this note we show that any graph of n vertices which is the union of two comparability graphs on the same vertex set, contains either a clique or an independent set of size at least n 13 . On the other hand, there exist such graphs for w...

We show that the problem of deciding whether a digraph has a Hamiltonian path between two specified vertices and the problem of deciding whether a given graph has a cubic subgraph are complete for monotone NP via monotone projection translations. We also show that the problem of deciding whether a uniquely partially orderable (resp. comparability) graph has a cubic subgraph is complete for NP v...

In this paper, at rst we describe a graph representing all the weak-order extensions of a partially ordered set and an algorithm generating them. Then we present a graph representing all of the minimal weak-order extensions of a partially ordered set, and implying a generation algorithm. Finally, we prove that the number of weak-order extensions of a partially ordered set is a comparability inv...

Similarities induced by a comparability mapping : meaning and utility in the context of the clustering of comparable texts. In the presence of bilingual comparable corpora it is natural to embed the data in two distinct linguistic representation spaces in which a "computational" notion of similarity is potentially defined. As far as these bilingual data are comparable in the sense of a measure ...

We show that if a tolerance graph is the complement of a comparability graph it is a trapezoid graph, i.e., the complement of an order of interval dimension at most 2. As consequences we are able to give obstructions for the class of bounded tolerance graphs and to give an example of a graph which is alternatingly orientable but not a tolerance graph. We also characterize the tolerance graphs a...