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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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