Quasi‐transitive digraphs

Abstract

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interesting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, w e show that every strongly connected quasi-transitive digraph D on at least four vertices has two vertices u1 and u2 such that D -u, is strongly connected for i = 1,2. A result of tournaments on the existence of a pair of arc-disjoint in-and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore w e characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 I k I IV(D)l -1 through every vertex and yet they are not hamiltonian. Finally w e characterize pancyclic and vertex pancyclic quasi-transitive digraphs. 0 1995, John Wiley & Sons, Inc.

Generalizations of tournaments: A survey

“…Bang-Jensen and Huang [25] reformulated this theorem (see below) and, using the result, found characterizations of pancyclic and vertex pancyclic quasi-transitive digraphs. They introduced the following notion.…”

“…DIGRAPHS AND THEIR GENERALIZATIONS Bang-Jensen and Huang [25] characterized both quasi-transitive digraphs containing hamiltonian cycles and hamiltonian paths (Theorem 8.1) using the analogues of Theorems 7.7 and 7.13 for extended semicomplete digraphs as well as Theorem 6.1. Bang-Jensen and Huang [25] Let Φ 0 be the union of all semicomplete multipartite, extended locally semicomplete and acyclic digraphs, Φ 1 be the union of all extended locally semicomplete and acyclic digraphs, and Φ 2 be the union of all semicomplete bipartite, extended locally semicomplete and acyclic digraphs.…”

“…Bang-Jensen and Huang [25] Let Φ 0 be the union of all semicomplete multipartite, extended locally semicomplete and acyclic digraphs, Φ 1 be the union of all extended locally semicomplete and acyclic digraphs, and Φ 2 be the union of all semicomplete bipartite, extended locally semicomplete and acyclic digraphs. It was proved [13,14] …”

Disjoint quasi‐kernels in digraphs

“…Our proof will mirror the following characterization of quasi-transitive digraphs, cf. [4]. We begin with strong quasi-transitive digraphs.…”

A note on vertex pancyclic oriented graphs

“…There are not many results about vertex pancyclicity in digraphs [2,4,11]. However, in many articles, various degree conditions have been obtained for digraphs to be Hamiltonian or pancyclic (see, e.g., [5,6,[8][9][10]).…”