Generalised irredundance in graphs: Nordhaus-Gaddum bounds

For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s,S), q = q(s,S) and r = r(s,S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p,q,r) may be considered as a compound existence property of S-pns. The subset S is called an f-set of G if f = 1 for all s ∈ S and the class of f-sets of G is denoted by ${\displaystyle {\Omega }_{f\left(G\right)}}$. Only 64 Boolean functions f can produce different classes ${\displaystyle {\Omega }_{f\left(G\right)}}$, special cases of which include the independent sets, irredundant sets, open irredundant sets and CO-irredundant sets of G. Let ${\displaystyle Q_{f\left(G\right)}}$ be the maximum cardinality of an f-set of G. For each of the 64 functions f, we establish sharp upper bounds for the sum ${\displaystyle Q_{f\left(G\right)}+Q_{f(G̅)}}$ and the product ${\displaystyle Q_{f\left(G\right)}{Q}_{f(G̅)}}$ in terms of n, the order of G.