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 Ωf (G). Only 64 Boolean functions f can produce different classes Ωf (G), special cases of which include the independent sets, irredundant sets, open irredundant sets and CO-irredundant sets of G. Let Qf (G) 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 Qf (G) + Qf (G) and the product Qf (G)Qf (G) in terms of n, the order of G.