On double domination in graphs

• Autorzy: Jochen Harant , Michael A. Henning
• 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 $γ_{×2}(G) ≤ ((ln(1+d) + lnδ + 1)/δ)n$.

Słowa kluczowe

EN

average degree; bounds; double domination; probabilistic method

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2005
• Tom: 25
• Numer: 1-2
• Strony: 29-34

Daty

wydano

2005

otrzymano

2003-10-22

poprawiono

2004-05-06

Twórcy

autor

Jochen Harant

• Department of Mathematics, Technical University of Ilmenau, D-98684 Ilmenau Germany

autor

Michael A. Henning

• 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.

Identyfikatory

DOI

10.7151/dmgt.1256