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2005 | 25 | 1-2 | 29-34
Tytuł artykułu

On double domination in graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{×2}(G)$. A function f(p) is defined, and it is shown that $γ_{×2}(G) = min f(p)$, where the minimum is taken over the n-dimensional cube $Cⁿ = {p = (p₁,...,pₙ) | p_i ∈ IR, 0 ≤ p_i ≤ 1,i = 1,...,n}$. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then $γ_{×2}(G) ≤ ((ln(1+d) + lnδ + 1)/δ)n$.
Wydawca
Rocznik
Tom
25
Numer
1-2
Strony
29-34
Opis fizyczny
Daty
wydano
2005
otrzymano
2003-10-22
poprawiono
2004-05-06
Twórcy
  • Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau Germany
  • School of Mathematics, Statistics, &, Information Technology, University of KwaZulu-Natal, Pietermaritzburg, 3209 South Africa
Bibliografia
  • [1] M. Blidia, M. Chellali and T.W. Haynes, Characterizations of trees with equal paired and double domination numbers, submitted for publication.
  • [2] M. Blidia, M. Chellali, T.W. Haynes and M.A. Henning, Independent and double domination in trees, submitted for publication.
  • [3] M. Chellali and T.W. Haynes, Paired and double domination in graphs, Utilitas Math., to appear.
  • [4] J. Harant, A. Pruchnewski and M. Voigt, On dominating sets and independent sets of graphs, Combin. Prob. and Comput. 8 (1998) 547-553, doi: 10.1017/S0963548399004034.
  • [5] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213.
  • [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
  • [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998).
  • [8] M.A. Henning, Graphs with large double domination numbers, submitted for publication.
  • [9] C.S. Liao and G.J. Chang, Algorithmic aspects of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002) 415-420.
  • [10] C.S. Liao and G.J. Chang, k-tuple domination in graphs, Information Processing Letters 87 (2003) 45-50, doi: 10.1016/S0020-0190(03)00233-3.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1256
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