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The second neighborhood conjecture of Seymour asserts that for any orientation G = (V, E), there exists a vertex v ∈ V so that |N + (v)| ≤ |N ++ (v)|. The conjecture was resolved by Fisher for tournaments. In this paper we prove the second neighborhood conjecture for several additional classes of dense orientations. We also prove some approximation results,(More)
A k-majority tournament is realized by 2k −1 linear orders on the set of vertices, where a vertex u dominates v if u precedes v in at least k of the orders. Various properties of such tournaments have been studied, among them the problem of finding the size of a smallest dominating set. It is known that 2-majority tournaments are dominated by 3 vertices and(More)
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