Generalised irredundance in graphs: Nordhaus-Gaddum bounds

• Volume: 24, Issue: 1, page 147-160
• ISSN: 2083-5892

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Abstract

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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 ${\Omega }_{f}\left(G\right)$. Only 64 Boolean functions f can produce different classes ${\Omega }_{f}\left(G\right)$, special cases of which include the independent sets, irredundant sets, open irredundant sets and CO-irredundant sets of G. Let ${Q}_{f}\left(G\right)$ 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 ${Q}_{f}\left(G\right)+{Q}_{f}\left(G̅\right)$ and the product ${Q}_{f}\left(G\right){Q}_{f}\left(G̅\right)$ in terms of n, the order of G.

How to cite

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Ernest J. Cockayne, and Stephen Finbow. "Generalised irredundance in graphs: Nordhaus-Gaddum bounds." Discussiones Mathematicae Graph Theory 24.1 (2004): 147-160. <http://eudml.org/doc/270673>.

@article{ErnestJ2004,
abstract = {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.},
author = {Ernest J. Cockayne, Stephen Finbow},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph; generalised irredundance; Nordhaus-Gaddum; irredundant; Boolean function},
language = {eng},
number = {1},
pages = {147-160},
title = {Generalised irredundance in graphs: Nordhaus-Gaddum bounds},
url = {http://eudml.org/doc/270673},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Ernest J. Cockayne
AU - Stephen Finbow
TI - Generalised irredundance in graphs: Nordhaus-Gaddum bounds
JO - Discussiones Mathematicae Graph Theory
PY - 2004
VL - 24
IS - 1
SP - 147
EP - 160
AB - 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.
LA - eng
KW - graph; generalised irredundance; Nordhaus-Gaddum; irredundant; Boolean function
UR - http://eudml.org/doc/270673
ER -

References

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