The minimum number of vertices in uniform hypergraphs with given domination number

Abstract The domination number γ ( H ) of a hypergraph H = ( V ( H ) , E ( H ) ) is the minimum size of a subset D ⊂ V ( H ) of the vertices such that for every v ∈ V ( H ) ∖ D there exist a vertex d ∈ D and an edge H ∈ E ( H ) with v , d ∈ H . We address the problem of finding the minimum number n ( k , γ ) of vertices that a k -uniform hypergraph H can have if γ ( H ) ≥ γ and H does not contain isolated vertices. We prove that n ( k , γ ) = k + Θ ( k 1 − 1 ∕ γ ) and also consider the s -wise dominating and the distance- l dominating version of the problem. In particular, we show that the minimum number n d c ( k , γ , l ) of vertices that a connected k -uniform hypergraph with distance- l domination number γ can have isroughly k γ l 2 .