Generalised irredundance in graphs: Nordhaus-Gaddum bounds

Ernest J. Cockayne; Stephen Finbow

Discussiones Mathematicae Graph Theory, Tome 24 (2004), p. 147-160 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $Q_{f} (G) + Q_{f} (G ̅)$ and the product $Q_{f} (G) Q_{f} (G ̅)$ in terms of n, the order of G.

Publié le : 2004-01-01

Zbl 1055.05111

EUDML-ID : urn:eudml:doc:270673

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Ernest J. Cockayne; Stephen Finbow. Generalised irredundance in graphs: Nordhaus-Gaddum bounds. Discussiones Mathematicae Graph Theory, Tome 24 (2004) pp. 147-160. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1221/

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