Geodesic
The Second Neighbourhood for Bipartite Tournaments
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 555-565

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Let T (X ∪ Y, A) be a bipartite tournament with partite sets X, Y and arc set A. For any vertex x ∈ X ∪Y, the second out-neighbourhood N++(x) of x is the set of all vertices with distance 2 from x. In this paper, we prove that T contains at least two vertices x such that |N++(x)| ≥ |N+(x)| unless T is in a special class ℬ1 of bipartite tournaments; show that T contains at least a vertex x such that |N++(x)| ≥ |N−(x)| and characterize the class ℬ2 of bipartite tournaments in which there exists exactly one vertex x with this property; and prove that if |X| = |Y | or |X| ≥ 4|Y |, then the bipartite tournament T contains a vertex x such that |N++(x)|+|N+(x)| ≥ 2|N−(x)|.
Keywords: second out-neighbourhood, out-neighbourhood, in-neighbourhood, bipartite tournament 
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     title = {The {Second} {Neighbourhood} for {Bipartite} {Tournaments}},
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Li, Ruijuan; Sheng, Bin. The Second Neighbourhood for Bipartite Tournaments. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 555-565. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a17/