An inequality chain of domination parameters for trees

• Autorzy: E.J. Cockayne , O. Favaron , J. Puech , C.M. Mynhardt
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

domination; irredundance; packing; perfect neighbourhoods; annihilation

Kategorie tematyczne

• 05C05: Trees
• 05C70: Factorization, matching, partitioning, covering and packing

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1998
• Tom: 18
• Numer: 1
• Strony: 127-142

Daty

wydano

1998

otrzymano

1997-10-30

Twórcy

autor

E.J. Cockayne

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

autor

O. Favaron

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

autor

J. Puech

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

autor

C.M. Mynhardt

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

Bibliografia

• [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).
• [6] E.J. Cockayne and C.M. Mynhardt, On a conjecture concerning irredundant and perfect neighbourhood sets in graphs, J. Combin. Math. Combin. Comput. (to appear).
• [7] O. Favaron and J. Puech, Irredundant and perfect neighbourhood sets in graphs and claw-free graphs, Discrete Math. (to appear).
• [8] G.H. Fricke, T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi and M.A. Henning, Perfect neighborhoods in graphs, (preprint).
• [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.

Identyfikatory

DOI

10.7151/dmgt.1069