Total Domination in Partitioned Graphs

We present results on total domination in a partitioned graph G = (V,E). Let γt(G) denote the total dominating number of G. For a partition V1, V2, . . . , Vk, k ≥ 2, of V , let γt(G;Vi) be the cardinality of a smallest subset of V such that every vertex of Vi has a neighbour in it and define the following ft(G;V1, V2, . . . , Vk) = γt(G) + γt(G;V1) + γt(G;V2) + . . .+ γt(G;Vk) ft(G; k) = max{ft(G;V1, V2, . . . , Vk) | V1, V2, . . . , Vk is a partition of V } gt(G; k) = max{Σk i=1γt(G;Vi) | V1, V2, . . . , Vk is a partition of V } We summarize known bounds on γt(G) and for graphs with all degrees at least δ we derive the following bounds for ft(G; k) and gt(G; k). (i) For δ ≥ 2 and k ≥ 3 we prove ft(G; k) ≤ 11|V |/7 and this inequality is best possible. (ii) for δ ≥ 3 we prove that ft(G; 2) ≤ (5/4 − 1/372)|V |. That inequality may not be best possible, but we conjecture that ft(G; 2) ≤ 7|V |/6 is. (iii) for δ ≥ 3 we prove ft(G; k) ≤ 3|V |/2 and this inequality is best possible. (iv) for δ ≥ 3 the inequality gt(G; k) ≤ 3|V |/4 holds and is best possible.