Improving some bounds for dominating Cartesian products

Bert L. Hartnell; Douglas F. Rall

Discussiones Mathematicae Graph Theory (2003)

  • Volume: 23, Issue: 2, page 261-272
  • ISSN: 2083-5892

Abstract

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The study of domination in Cartesian products has received its main motivation from attempts to settle a conjecture made by V.G. Vizing in 1968. He conjectured that γ(G)γ(H) is a lower bound for the domination number of the Cartesian product of any two graphs G and H. Most of the progress on settling this conjecture has been limited to verifying the conjectured lower bound if one of the graphs has a certain structural property. In addition, a number of authors have established bounds for dominating the Cartesian product of any two graphs. We show how it is possible to improve some of these bounds by imposing conditions on both graphs. For example, we establish a new lower bound for the domination number of T T, when T is a tree, and we improve an upper bound of Vizing in the case when one of the graphs has k > 1 dominating sets which cover the vertex set and the other has a dominating set which partitions in a certain way.

How to cite

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Bert L. Hartnell, and Douglas F. Rall. "Improving some bounds for dominating Cartesian products." Discussiones Mathematicae Graph Theory 23.2 (2003): 261-272. <http://eudml.org/doc/270360>.

@article{BertL2003,
abstract = {The study of domination in Cartesian products has received its main motivation from attempts to settle a conjecture made by V.G. Vizing in 1968. He conjectured that γ(G)γ(H) is a lower bound for the domination number of the Cartesian product of any two graphs G and H. Most of the progress on settling this conjecture has been limited to verifying the conjectured lower bound if one of the graphs has a certain structural property. In addition, a number of authors have established bounds for dominating the Cartesian product of any two graphs. We show how it is possible to improve some of these bounds by imposing conditions on both graphs. For example, we establish a new lower bound for the domination number of T T, when T is a tree, and we improve an upper bound of Vizing in the case when one of the graphs has k > 1 dominating sets which cover the vertex set and the other has a dominating set which partitions in a certain way.},
author = {Bert L. Hartnell, Douglas F. Rall},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination number; Cartesian product; Vizing's conjecture; 2-packing; domination},
language = {eng},
number = {2},
pages = {261-272},
title = {Improving some bounds for dominating Cartesian products},
url = {http://eudml.org/doc/270360},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Bert L. Hartnell
AU - Douglas F. Rall
TI - Improving some bounds for dominating Cartesian products
JO - Discussiones Mathematicae Graph Theory
PY - 2003
VL - 23
IS - 2
SP - 261
EP - 272
AB - The study of domination in Cartesian products has received its main motivation from attempts to settle a conjecture made by V.G. Vizing in 1968. He conjectured that γ(G)γ(H) is a lower bound for the domination number of the Cartesian product of any two graphs G and H. Most of the progress on settling this conjecture has been limited to verifying the conjectured lower bound if one of the graphs has a certain structural property. In addition, a number of authors have established bounds for dominating the Cartesian product of any two graphs. We show how it is possible to improve some of these bounds by imposing conditions on both graphs. For example, we establish a new lower bound for the domination number of T T, when T is a tree, and we improve an upper bound of Vizing in the case when one of the graphs has k > 1 dominating sets which cover the vertex set and the other has a dominating set which partitions in a certain way.
LA - eng
KW - domination number; Cartesian product; Vizing's conjecture; 2-packing; domination
UR - http://eudml.org/doc/270360
ER -

References

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  1. [1] A.M. Barcalkin and L.F. German, The external stability number of the Cartesian product of graphs, Bul. Akad. Stiince RSS Moldoven. 1 (1979) 5-8. 
  2. [2] W.E. Clark and S. Suen, An inequality related to Vizing's conjecture, Elec. J. Combin. 7 (#N4) (2000) 1-3. Zbl0947.05056
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  4. [4] B.L. Hartnell, On determining the 2-packing and domination numbers of the Cartesian product of certain graphs, Ars Combin. 55 (2000) 25-31. Zbl0994.05102
  5. [5] B.L. Hartnell and D.F. Rall, On Vizing's conjecture, Congr. Numer. 82 (1991) 87-96. Zbl0764.05092
  6. [6] B.L. Hartnell and D.F. Rall, Chapter 7: Domination in Cartesian products: Vizing's conjecture, in: T.W. Haynes, S.T. Hedetniemi and P.J. Slater, eds, Domination in Graphs: Advanced Topics, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 209 (Marcel Dekker, Inc., New York, 1998). Zbl0890.05035
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  8. [8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 208 (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
  9. [9] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079. Zbl0602.05043
  10. [10] M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs: I, Ars Combin. 18 (1983) 33-44. Zbl0566.05050
  11. [11] M.S. Jacobson and L.F. Kinch, On the domination of the products of graphs II: trees, J. Graph Theory 10 (1986) 97-106, doi: 10.1002/jgt.3190100112. Zbl0584.05053
  12. [12] V.G. Vizing, The Cartesian product of graphs, Vy cisl. Sistemy 9 (1963) 30-43. 
  13. [13] V.G. Vizing, Some unsolved problems in graph theory, Uspehi Mat. Nauk 23 (6) (1968) 117-134. Zbl0177.52301

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