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2003 | 23 | 2 | 261-272
Tytuł artykułu

Improving some bounds for dominating Cartesian products

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
23
Numer
2
Strony
261-272
Opis fizyczny
Daty
wydano
2003
otrzymano
2001-10-01
poprawiono
2002-01-20
Twórcy
  • Saint Mary's University, Halifax, Nova Scotia, Canada B3H 3C3
  • Furman University, Greenville, SC 29613, USA
Bibliografia
  • [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] W.E. Clark and S. Suen, An inequality related to Vizing's conjecture, Elec. J. Combin. 7 (#N4) (2000) 1-3.
  • [3] M. El-Zahar and C.M. Pareek, Domination number of products of graphs, Ars Combin. 31 (1991) 223-227.
  • [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.
  • [5] B.L. Hartnell and D.F. Rall, On Vizing's conjecture, Congr. Numer. 82 (1991) 87-96.
  • [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).
  • [7] B.L. Hartnell and D.F. Rall, Lower bounds for dominating Cartesian products, J. Combin. Math. Combin. Comp. 31 (1999) 219-226.
  • [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).
  • [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.
  • [10] M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs: I, Ars Combin. 18 (1983) 33-44.
  • [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.
  • [12] V.G. Vizing, The Cartesian product of graphs, Vy cisl. Sistemy 9 (1963) 30-43.
  • [13] V.G. Vizing, Some unsolved problems in graph theory, Uspehi Mat. Nauk 23 (6) (1968) 117-134.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1201
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