Multiple factor NordhausGaddum type results for domination and total domination

A NordhausGaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we examine the sum and product of ^γt( ^G1), ^γt( ^G2),..., ^γt( ^Gk) and the sum of γ( ^G1),γ( ^G2),...,γ( ^Gk) where ^G1⊕ ^G2⊕⋯⊕ ^Gk= ^Kn for positive integers n and k, γ(G) is the domination number and ^γt(G) is total domination number of a graph G. We show that ∑j=1kγ( ^Gj)≤(k-1)n+1 with equality if and only if ^Gi= ^Kn for some i∈1,...,k. For n≥7, 3≤k≤n-2 and δ( ^Gi)≥1 for each i∈1,2,...,k, we show that ∑j=1k ^γt( ^Gj)≤(k-1)(n+1).