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Paths, Stars and the Number Three

Published online by Cambridge University Press:  12 September 2008

Bruce Reed
Bruce Reed
Affiliation:
Equipe Combinatoire, Institute Blaise Pascal, Université Paris VI, 4 Place Jussieu, Paris 75252, Cedex 05, France Email: [email protected]
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Abstract

A dominating set for a graph G is a set D of vertices of G such that every vertex of G not in D is adjacent to a vertex of D. We prove that any graph G of minimum degree at least three contains a dominating set D of size at most 3|V(G)|/8. A star S is a graph consisting of a centre x and a set of edges from x to S — x. Clearly, a dominating set D for a graph G corresponds to a set of |D| stars which cover V(G). Thus, we show that the vertices of any graph G of minimum degree 3 can be covered by at most 3|V(G)|/8 vertex disjoint stars. We also show that any connected cubic graph G can be covered by [|V(G)|/9] vertex disjoint paths. Both these results are sharp.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 3 , September 1996 , pp. 277 - 295
Copyright
Copyright © Cambridge University Press 1996

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References

[1]Arnautov, V. I. (1974) Estimation of the external stability number of a graph by means of the minimal degree of the vertices. Prikl. Mat. i Programmirovanie Vyp. 11 3–8.Google Scholar
[2]Blank, M. (1973) An estimate of the external stability number of a graph without suspended vertices. Prikl. Math, i Programmirovanie Vyp. 10 3–11.Google Scholar
[3]Cockayne, E. J. and Hedetnemie, S. T. (1977) Towards a theory of domination in graphs. Networks 7 241–261.CrossRefGoogle Scholar
[4]Edmonds, J. (1964) Paths, trees and flowers. Can. J. Math. 17 449–467.CrossRefGoogle Scholar
[5]Lovász, L. (1975) On the ratio of optimal integral and fractional covers. Discrete Mathematics 13 383–390.CrossRefGoogle Scholar
[6]McCuaig, B. and Shepherd, B. (1989) Domination in graphs of minimum degree two. Journal of Graph Theory 13 749–762.CrossRefGoogle Scholar
[7]Payan, C. (1975) Sur la Nombre d'Absorption d'un Graphe Simple. Cahiers C.E.R.O. 17 307–317.Google Scholar
[8]Reed, B. (1993) Paths, stars and the number three: the grunge. University of Waterloo, Technical Report.Google Scholar