DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory  18(1) (1998)   127-142
DOI: 10.7151/dmgt.1069

AN INEQUALITY CHAIN OF DOMINATION PARAMETERS FOR TREES

E.J. Cockayne

Department of Mathematics, University of Victoria
Victoria, British Columbia, Canada V8W 3P4

e-mail: cockayne@smart.math.uvic.ca

O. Favaron and J. Puech

LRI, Bât. 490, Université de Paris-Sud
Orsay, France 91405

e-mail: of@lri.fre-mail: puech@lri.fr

C.M. Mynhardt

Department of Mathematics, University of South Africa
PO Box 392, 0003 Pretoria, South Africa

e-mail: mynhacm@alpha.unisa.ac.za

Abstract

We prove that the smallest cardinality of a maximal packing in any tree is at most the cardinality of an R-annihilated set. As a corollary to this result we point out that a set of parameters of trees involving packing, perfect neighbourhood, R-annihilated, irredundant and dominating sets is totally ordered. The class of trees for which all these parameters are equal is described and we give an example of a tree in which most of them are distinct.

Keywords: domination, irredundance, packing, perfect neighbourhoods, annihilation.

1991 Mathematics Subject Classification: 05C70, 05C05.

References

[1] E.J. Cockayne, O. Favaron, C.M. Mynhardt and J. Puech, A characterisation of (γ,i)-trees, (preprint).
[2] E.J. Cockayne, O. Favaron, C.M. Mynhardt and J. Puech, Packing, perfect neighbourhood, irredundant and R-annihilated sets in graphs, Austr. J. Combin. Math. (to appear).
[3] E.J. Cockayne, P.J. Grobler, S.T. Hedetniemi and A.A. McRae, What makes an irredundant set maximal ?, J. Combin. Math. Combin. Comput. (to appear).
[4] E.J. Cockayne, J.H. Hattingh, S.M. Hedetniemi, S.T. Hedetniemi and A.A. McRae, Using maximality and minimality conditions to construct inequality chains, Discrete Math. 176 (1997) 43-61, doi: 10.1016/S0012-365X(96)00356-1.
[5] E.J. Cockayne, S.M. Hedetniemi, S.T. Hedetniemi and C.M. Mynhardt, Irredundant and perfect neighbourhood sets in trees, Discrete Math. (to appear).
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[7] O. Favaron and J. Puech, Irredundant and perfect neighbourhood sets in graphs and claw-free graphs, Discrete Math. (to appear).
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[9] B.L. Hartnell, On maximal radius two independent sets, Congr. Numer. 48 (1985) 179-182.
[10] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs (Marcel Dekker, 1997).
[11] A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1975) 225-233.
[12] J. Puech, Irredundant and independent perfect neighborhood sets in graphs, (preprint).
[13] J. Topp and L. Volkmann, On packing and covering numbers of graphs, Discrete Math. 96 (1991) 229-238, doi: 10.1016/0012-365X(91)90316-T.

Received 30 October 1997