Total domination in partitioned trees and partitioned graphs with minimum degree two

Let G = (V, E) be a graph and let S ⊆ V . A set of vertices in G totally dominates S if every vertex in S is adjacent to some vertex of that set. The least number of vertices needed in G to totally dominate S is denoted by γ _t (G, S). When S = V, γ _t (G, V) is the well studied total domination number γ _t (G). We wish to maximize the sum γ _t (G) + γ _t (G, V ₁) + γ _t (G, V ₂) over all possible partitions V ₁, V ₂ of V. We call this maximum sum f _t (G). For a graph H, we denote by H ^ P ₂ the graph obtained from H by attaching a path of length 2 to each vertex of H so that the resulting paths are vertex-disjoint. We show that if G is a tree of order n ≥ 4 and G ∉ {P₅, P₆, P₇, P₁₀, P₁₄, then f _t (G) ≤ 14n/9 with equality if and only if G {P ₉, P ₁₈} or G = (T ^P ₂) ^P ₂ for some tree T. If G is a connected graph of order n with minimum degree at least two, we establish that f _t (G) ≤ 3n/2 with equality if and only if G is a cycle of order congruent to zero modulo 4.