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 γt(G,S). When S = V , γt(G,V ) is the well studied total domination number γt(G). We wish to maximize the sum γt(G) + γt(G,V1) + γt(G,V2) over all possible partitions V1, V2 of V . We call this maximum sum ft(G). For a graph H, we denote by H ◦ P2 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 ft(G) ≤ 14n/9 with equality if and only if G ∈ {P9, P18} or G = (T ◦ P2) ◦ P2 for some tree T . If G is a connected graph of order n with minimum degree at least two, we establish that ft(G) ≤ 3n/2 with equality if and only if G is a cycle of order congruent to zero modulo 4.