Total domination in partitioned graphs

We present results on total domination in a partitioned graph G = (V, E). Let γ _t (G) denote the total dominating number of G. For a partition V₁, V₂, , V_k, k ≥2, of V, let γ _t (G; V _i ) be the cardinality of a smallest subset of V such that every vertex of V _i has a neighbour in it and define the following f_t(G; V₁, V₂, , V _k) = γ _t (G) + γ _t (G; V ₁) + γ _t (G; V₂) ++γ_t(G; V_k) f_t (G; k) = max f_t (G; V₁, V₂, , V_k) mid V₁, V₂, , V _k rm is a partition of V g_t (G; k) = max Σ _i=1^kγ_t (G; V_i) \mid V ₁, V₂, l, V_k rm is a partition of V We summarize known bounds on γ _t (G) and for graphs with all degrees at least δ we derive the following bounds for f _t (G; k) and g _t (G; k). (i) For δ ≥2 and k ≥ 3 we prove f _t (G; k) ≤ 11|V|/7 and this inequality is best possible. (ii) for δ ≤3 we prove that f _t (G; 2) ≤ (5/4 - 1/372)|V|. That inequality may not be best possible, but we conjecture that f _t (G; 2) ≤ 7|V|/6 is. (iii) for δ 3 we prove f _t (G; k) ≤3|V|/2 and this inequality is best possible. (iv) for δ ≥3 the inequality g _t (G; k) ≥3|V|/4 holds and is best possible.