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On transitive orientations of \(G-\widehat e\). (English) Zbl 1193.05140
Summary: A comparability graph is a graph whose edges can be oriented transitively. Given a comparability graph \(G=(V,E)\) and an arbitrary edge \(\widehat e\in E\) we explore the question whether the graph \(G-\widehat e\), obtained by removing the undirected edge \(\widehat e\), is a comparability graph as well. We define a new substructure of implication classes and present a complete mathematical characterization of all those edges.
MSC:
05C75 Structural characterization of families of graphs
05C20 Directed graphs (digraphs), tournaments
90B35 Deterministic scheduling theory in operations research
68R10 Graph theory (including graph drawing) in computer science
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