The structure of strong arc-locally in-semicomplete digraphs
The main result in [\textit{J. Bang-Jensen}, ``The structure of strong arc-locally semicomplete digraphs,'' Discrete Math. 283, No. 1--3, 1--6 (2004; Zbl 1048.05038)] is that a strong arc-locally semicomplete digraph is either semicomplete, semicomplete bipartite or an extended cycle. The authors show that in this classification a class of digraphs that they call \(W_n\) must be included. The goal of the paper is to obtain an analogous result for strong arc-locally in-semicomplete digraphs. It is shown that a strong arc-locally in-semicomplete digraph is either semicomplete, semicomplete bipartite, an extended cycle or a \(T\)-digraph. Roughly speaking, a \(T\)-digraph is defined as a digraph formed by four pieces (two digraphs with no arcs, a trivial digraph, and a semicomplete digraph) and some adjacencies between these pieces. The result is used to show that a conjecture of Bang-Jensen is true (Conjecture 4.3 in the above mentioned reference).
scientific article; zbMATH DE number 5648216