On weak and restrained domination in trees

Abstract

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)  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 l leaves and s support vertices, then γw(T), γr(T)  ⌈(n + 2 + l – s)/3⌉, and γw(T), γr(T) ≥ γ(T) + l – s ≥ ⌈(n + 2 + 2l – 3s)/3⌉ improving those of Hattingh and Rautenbach. We also show that γw(T)  ⌋(n + 2l + 2s – 3)/3⌊ and γr(T)  ⌋(n + 2l + s + 1)/3⌊.