Let H be a 3-uniform hypergraph of order n and size m and let T be a subset of vertices of H. The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number τ s (H) of H is the minimum size of a strong transversal in H. We show that 7τ s (H) ≤ 4n+2m and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph H of order n ≥ 6 and size m that achieves equality in this bound. A set S of vertices in a graph G is a double total dominating set of G if every vertex of G is adjacent to at least two vertices in S. The minimum cardinality of a double total dominating set of G is the double total domination number γ ×2,t (G) of G. Let G be a connected graph of order n with minimum degree at least three. Using our hypergraph results, we show that γ ×2,t (G) ≤ 6n/7 with equality if and only if G is the Heawood graph.
Theorem 1.1. REF If G = G 14 is a connected graph of order n with δ(G) ≥ 3, then γ 2,2 (G) ≤