On the Total Positive Influence Domination in Graphs

Let G be a graph without isolated vertices. A set D⊆ V(G) is a total positive influence dominating set of G if |NG(v)∩ D|≥|NG(v)|/2 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 article we obtain tight bounds on γₜ⁺(G) in terms of several parameters of G, and we characterize the graphs achieving some of them. In particular, we show that γₜ⁺(G)≤ 8n/9 for every graph G of order n with δ(G)≥ 2, and that γₜ⁺(G)≤ 5n/6 for every connected graph G of order n with 2≤δ(G)≤Δ(G)≤ 3 different from the Heawood graph. Finally, we study the behavior of γₜ⁺(G) under edge deletion.