Total positive influence domination number of trees

Let G be a graph with no isolated vertex. A set D⊆ V(G) is a total positive influence dominating set of G if |NG(v)∩ D|≥ |NG(v) D| for every vertex v∈ V(G), where NG(v) represents the open neighborhood of v. The total positive influence domination number of G, denoted by γₜ⁺(G), is the minimum cardinality among all total positive influence dominating sets of G. In this paper we study this parameter in trees. In particular, we obtain tight lower bounds on γₜ⁺(T) in terms of the number of leaves and the number of vertices of even degree of a tree T, and we characterize the trees attaining them. Moreover, we show that γₜ⁺(T)≤ 3n/4 for every tree T of order n≥ 3, and we characterize the trees for which the equality holds. We also obtain tight bounds on γₜ⁺(T) in terms of the number of leaves, the support vertices and the independence number of T. Finally, we show that n/2≤ γₜ⁺(T̄)≤ n/2+1 for every tree T of order n different from a star, and as a consequence, we obtain Nordhaus--Gaddum type inequalities for this parameter.