Geodesic
Generalised irredundance in graphs: Nordhaus-Gaddum bounds
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 1, pp. 147-160

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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 Ω_f(G). Only 64 Boolean functions f can produce different classes Ω_f(G), special cases of which include the independent sets, irredundant sets, open irredundant sets and CO-irredundant sets of G. Let Q_f(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 Q_f(G) + Q_f(G̅) and the product Q_f(G)Q_f(G̅) in terms of n, the order of G.
Keywords: graph, generalised irredundance, Nordhaus-Gaddum 
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Cockayne, Ernest; Finbow, Stephen. Generalised irredundance in graphs: Nordhaus-Gaddum bounds. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 1, pp. 147-160. http://geodesic.mathdoc.fr/item/DMGT_2004_24_1_a12/