On weak and restrained domination in trees

In a graph G = (V,E) a vertex is said to dominate itself and all its neighbours. A weak dominating set is a set S ⊆ V where for every vertex u not in S there is a vertex v of S adjacent to u with dG(v) 6 dG(u) . A restrained dominating set is a set S ⊆ V where every vertex in V − S is adjacent to a vertex in S as well as another vertex in V − S . The weak domination number γw(G) (resp. restrained domination number γr(G)) is the minimum cardinality of a weak dominating set (resp. restrained dominating set). We determine sharp bounds for the weak and restrained domination numbers of a tree in terms of the domination number, the order, number of leaves and support vertices. More precisely, we show that if T is a tree of order n ≥ 3 with ` leaves and s support vertices, then γw(T ), γr(T ) > d(n + 2 + `− s)/3e , and γw(T ), γr(T ) ≥ γ(T ) + ` − s ≥ d(n + 2 + 2`− 3s)/3e improving those of Hattingh and Rautenbach. We also show that γw(T ) 6 b(n + 2` + 2s− 3)/3c and γr(T ) 6 b(n + 2` + s + 1)/3c .