text: Let @math be a connected graph of order @math , whose minimum vertex degree is at least @math . A subset @math of vertices in @math is a @math -tuple total dominating set if every vertex of @math is adjacent to at least @math vertices in @math . The minimum cardinality of a @math -tuple total dominating set of @math is the @math -tuple total domination number of @math , denoted by @math . Henning and Yeo in hen proved that if @math is a cubic graph different from the Heawood graph, @math , and this bound is sharp. Similarly, a @math -tuple dominating set is a subset @math of vertices of @math , @math such that @math for every vertex @math , where @math . The @math -tuple domination number of @math , denoted by @math , is the minimum cardinality of a @math -tuple dominating set of @math . In this paper, we give a simple approach to compute an upper bound for @math -tuple total domination number of @math -regular graphs. Also, we give an upper bound for the @math -tuple dominating number of @math -regular graphs. In addition, our method gives algorithms to compute dominating sets with the given bounds, while the previous methods are existential.
  @cite_7: Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , 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 @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp.
summary: Let @math be the Heawood graph (or, equivalently, the incidence bipartite graph of the Fano plane) on @math vertices shown in Figure . In @cite_7 , Henning and Yeo proved some theorems about strong transversal in hypergraphs and then as an application of their hypergraph results they proved the following theorem.