DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 24(1) (2004) 147-160
DOI: https://doi.org/10.7151/dmgt.1221

GENERALISED IRREDUNDANCE IN GRAPHS: NORDHAUS-GADDUM BOUNDS

Ernest J. Cockayne and Stephen Finbow

University of Victoria
B.C., Canada V8W 3P4
e-mail: cockayne@math.uvic.ca

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 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.

Keywords: graph, generalised irredundance, Nordhaus-Gaddum.

2000 Mathematics Subject Classification: 05C69, 05C55.

References

[1] B. Bollobás and E.J. Cockayne, The irredundance number and maximum degree of a graph, Discrete Math. 69 (1984) 197-199. 
[2] E.J. Cockayne, Generalized irredundance in graphs: hereditary properties and Ramsey numbers, J. Combin. Math. Combin. Comput. 31 (1999) 15-31. 
[3] E.J. Cockayne, Nordhaus-Gaddum Results for Open Irredundance, J. Combin. Math. Combin. Comput., to appear. 
[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. 
[5] E.J. Cockayne, O. Favaron, C.M. Mynhardt, Open irredundance and maximum degree in graphs (submitted). 
[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. 
[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. 
[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. 
[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. 
[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. 
[12] E.J. Cockayne, C.M. Mynhardt, J. Simmons, The CO-irredundent Ramsey number t(4,7), Utilitas Math. 57 (2000) 193-209. 
[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. 
[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. 
[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. 
[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. 
[18] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). 
[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. 
[20] E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177, doi: 10.2307/2306658. 
[21] J. Simmons, CO-irredundant Ramsey numbers for graphs (Master's Thesis, University of Victoria, 1998). 

Received 20 March 2002
Revised 4 February 2003

Close