Abstract
In a simple digraph, a star of degree t is a union of t edges with a common tail. The k-domination number γ k(G) of digraph G is the minimum number of stars of degree at most k needed to cover the vertex set. We prove that γ k(T)=⌈n/(k+1)⌉ when T is a tournament with n⩾14k lg k vertices. This improves a result of Chen, Lu and West. We also give a short direct proof of the result of E. Szekeres and G. Szekeres that every n-vertex tournament is dominated by at most lg n− lg lg n+2 vertices.
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