## ON TRANSITIVE ORIENTATIONS OF G-ê

Michael Andresen

Otto-von-Guericke-Universität
Fakultät für Mathematik
PSF 4120, 39016 Magdeburg, Germany

## Abstract

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.

Keywords: (primary): comparability graph, edge deletion;
(secondary): transitive orientation, Triangle Lemma, Γ-components, open shop scheduling, irreducibility.

2000 Mathematics Subject Classification: 05C75, 05C20, 90B35, 68R10.

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Received 28 February 2007
Revised 18 February 2009
Accepted 18 February 2009