# Generalised irredundance in graphs: Nordhaus-Gaddum bounds

Ernest J. Cockayne; Stephen Finbow

Discussiones Mathematicae Graph Theory (2004)

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

## Access Full Article

top## Abstract

top## How to cite

topErnest 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

top- [1] B. Bollobás and E.J. Cockayne, The irredundance number and maximum degree of a graph, Discrete Math. 69 (1984) 197-199. Zbl0539.05056
- [2] E.J. Cockayne, Generalized irredundance in graphs: hereditary properties and Ramsey numbers, J. Combin. Math. Combin. Comput. 31 (1999) 15-31. Zbl0952.05051
- [3] E.J. Cockayne, Nordhaus-Gaddum Results for Open Irredundance, J. Combin. Math. Combin. Comput., to appear. Zbl1038.05042
- [4] E.J. Cockayne, O. Favaron, P.J.P. Grobler, C.M. Mynhardt and J. Puech, Ramsey properties of generalised irredundant sets in graphs, Discrete Math. 231 (2001) 123-134, doi: 10.1016/S0012-365X(00)00311-3. Zbl0977.05089
- [5] E.J. Cockayne, O. Favaron, C.M. Mynhardt, Open irredundance and maximum degree in graphs (submitted). Zbl1207.05095
- [6] E.J. Cockayne, P.J.P. Grobler, S.T. Hedetniemi and A.A. McRae, What makes an irredundant set maximal? J. Combin. Math. Combin. Comput. 25 (1997) 213-223. Zbl0907.05032
- [7] E.J. Cockayne, S.T. Hedetniemi, D.J. Miller, Properties of hereditary hypergraphs and middle graphs, Canad. Math. Bull. 21 (1978) 261-268, doi: 10.4153/CMB-1978-079-5. Zbl0393.05044
- [8] E.J. Cockayne, G. MacGillvray, J. Simmons, CO-irredundant Ramsey numbers for graphs, J. Graph Theory 34 (2000) 258-268, doi: 10.1002/1097-0118(200008)34:4<258::AID-JGT2>3.0.CO;2-4
- [9] E.J. Cockayne, D. McCrea, C.M. Mynhardt, Nordhaus-Gaddum results for CO-irredundance in graphs, Discrete Math. 211 (2000) 209-215, doi: 10.1016/S0012-365X(99)00282-4. Zbl0948.05037
- [10] E.J. Cockayne and C.M. Mynhardt, Irredundance and maximum degree in graphs, Combin. Prob. Comput. 6 (1997) 153-157, doi: 10.1017/S0963548396002891. Zbl0881.05068
- [11] E.J. Cockayne, C.M. Mynhardt, On the product of upper irredundance numbers of a graphs and its complement, Discrete Math. 76 (1988) 117-121, doi: 10.1016/0012-365X(89)90304-X. Zbl0672.05049
- [12] E.J. Cockayne, C.M. Mynhardt, J. Simmons, The CO-irredundent Ramsey number t(4,7), Utilitas Math. 57 (2000) 193-209. Zbl0956.05073
- [13] A.M. Farley and A. Proskurowski, Computing the maximum order of an open irredundant set in a tree, Congr. Numer. 41 (1984) 219-228. Zbl0546.05034
- [14] A.M. Farley and N. Schacham, Senders in broadcast networks: open irredundancy in graphs, Congr. Numer. 38 (1983) 47-57.
- [15] O. Favaron, A note on the open irredundance in a graph, Congr. Numer. 66 (1988) 316-318. Zbl0673.05083
- [16] O. Favaron, A note on the irredundance number after vertex deletion, Discrete Math. 121 (1993) 51-54, doi: 10.1016/0012-365X(93)90536-3. Zbl0791.05095
- [17] M.R. Fellows, G.H. Fricke, S.T. Hedetniemi and D. Jacobs, The private neighbour cube, SIAM J. Discrete Math. 7 (1994) 41-47, doi: 10.1137/S0895480191199026. Zbl0795.05078
- [18] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
- [19] S.T. Hedetniemi, D.P. Jacobs and R.C. Laskar, Inequalities involving the rank of a graph, J. Combin. Math. Combin. Comput. 6 (1989) 173-176. Zbl0732.05039
- [20] E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177, doi: 10.2307/2306658. Zbl0070.18503
- [21] J. Simmons, CO-irredundant Ramsey numbers for graphs (Master's Thesis, University of Victoria, 1998).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.