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Quasi-transitive digraphs. (English) Zbl 0832.05048
A digraph is quasi-transitive if whenever \(x\), \(y\) and \(z\) are distinct vertices such that \(x\) dominates \(y\) and \(y\) dominates \(z\), then \(x\) and \(z\) are adjacent. Quasi-transitive digraphs share many nice properties of tournaments. It is shown that every strongly connected quasi-transitive digraph \(D\) on \(n\geq 4\) vertices contains two vertices \(x\) and \(y\) such that \(D- x\) and \(D- y\) are strongly connected. Hamiltonian, pancyclic and vertex pancyclic quasi-transitive digraphs are characterized. The authors point out that their characterizations of quasi-transitive digraphs having hamiltonian cycle or path, respectively, do not seem to imply polynomial algorithms. Their conjecture that such algorithms exist has been confirmed by the reviewer [Australas. J. Comb. 10, 231-236 (1994; Zbl 0817.05061)].
Reviewer: G.Gutin (Odense)

MSC:
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C75 Structural characterization of families of graphs
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