On double domination in graphs

• Volume: 25, Issue: 1-2, page 29-34
• ISSN: 2083-5892

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Abstract

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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 ${\gamma }_{×2}\left(G\right)$. A function f(p) is defined, and it is shown that ${\gamma }_{×2}\left(G\right)=minf\left(p\right)$, where the minimum is taken over the n-dimensional cube $Cⁿ=p=\left(p₁,...,pₙ\right)|{p}_{i}\in IR,0\le {p}_{i}\le 1,i=1,...,n$. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then ${\gamma }_{×2}\left(G\right)\le \left(\left(ln\left(1+d\right)+ln\delta +1\right)/\delta \right)n$.

How to cite

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Jochen Harant, and Michael A. Henning. "On double domination in graphs." Discussiones Mathematicae Graph Theory 25.1-2 (2005): 29-34. <http://eudml.org/doc/270158>.

@article{JochenHarant2005,
abstract = {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 $γ_\{×2\}(G) ≤ ((ln(1+d) + lnδ + 1)/δ)n$.},
author = {Jochen Harant, Michael A. Henning},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {average degree; bounds; double domination; probabilistic method; dominating number},
language = {eng},
number = {1-2},
pages = {29-34},
title = {On double domination in graphs},
url = {http://eudml.org/doc/270158},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Jochen Harant
AU - Michael A. Henning
TI - On double domination in graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 1-2
SP - 29
EP - 34
AB - 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 $γ_{×2}(G) ≤ ((ln(1+d) + lnδ + 1)/δ)n$.
LA - eng
KW - average degree; bounds; double domination; probabilistic method; dominating number
UR - http://eudml.org/doc/270158
ER -

References

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7. [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998). Zbl0883.00011
8. [8] M.A. Henning, Graphs with large double domination numbers, submitted for publication.
9. [9] C.S. Liao and G.J. Chang, Algorithmic aspects of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002) 415-420. Zbl1047.05032
10. [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.

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