Some nordhaus‐‐ gaddum‐type results

A Nordhaus‐‐Gaddum‐type result is a (tgiht) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, the sums and products of ψ(G₁) and ψ(G₂) are examined where G₁ ⊕ G₂ = K(s, s), and ψ is the independence, domination, or independent domination number, inter alia. In particular, it is shown that the maximum value of the product of the domination numbers of G₁ and G₂ is [(s/2 + 2)²] for s ≥ 3. Thereafter it is shown that for H₁ ⊕ H₂ ⊕ H₃ = K_p, the maximum product of the domination numbers of H₂, H₂, and H₃ is p³/27 + Θ(p²).