Locating-domination, 2-domination and independence in trees

A set D of vertices in a graph G is 2-dominating if every vertex not in D has at least two neighbors in D and locating-dominating if for every two vertices u, v not in D, the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The minimum cardinality of a 2-dominating set (locatingdominating set) is denoted by γ2(G) (γL(G)). It is known that every tree T with n ≥ 2 vertices, leaves, s support vertices and independence number β(T ), satisfies γL(T ) ≤ (n + − s)/2 ≤ β(T ) ≤ γ2(T ). We show that β(T )+ γL(T ) ≤ n+ − s and that γ2(T ) = (n+ − s)/2 if and only if γL(T ) = γ2(T ). Moreover, γ2(G) ≤ 2γL(G) for every bipartite graph and γ2(G) ≤ 2γL(G)− 1 if G is a tree with n ≥ 3. 310 BLIDIA, FAVARON AND LOUNES