# Characterization of block graphs with equal 2-domination number and domination number plus one

• Volume: 27, Issue: 1, page 93-103
• ISSN: 2083-5892

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## Abstract

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Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)-D is adjacent with at least p vertices of D. The p-domination number γₚ(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination number γ(G). If G is a nontrivial connected block graph, then we show that γ₂(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ₂(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees.

## How to cite

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Adriana Hansberg, and Lutz Volkmann. "Characterization of block graphs with equal 2-domination number and domination number plus one." Discussiones Mathematicae Graph Theory 27.1 (2007): 93-103. <http://eudml.org/doc/270741>.

abstract = { Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)-D is adjacent with at least p vertices of D. The p-domination number γₚ(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination number γ(G). If G is a nontrivial connected block graph, then we show that γ₂(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ₂(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees. },
author = {Adriana Hansberg, Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination; 2-domination; multiple domination; block graph; block graph},
language = {eng},
number = {1},
pages = {93-103},
title = {Characterization of block graphs with equal 2-domination number and domination number plus one},
url = {http://eudml.org/doc/270741},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Lutz Volkmann
TI - Characterization of block graphs with equal 2-domination number and domination number plus one
JO - Discussiones Mathematicae Graph Theory
PY - 2007
VL - 27
IS - 1
SP - 93
EP - 103
AB - Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)-D is adjacent with at least p vertices of D. The p-domination number γₚ(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination number γ(G). If G is a nontrivial connected block graph, then we show that γ₂(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ₂(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees.
LA - eng
KW - domination; 2-domination; multiple domination; block graph; block graph
UR - http://eudml.org/doc/270741
ER -

## References

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10. [10] J. Topp and L. Volkmann, On domination and independence numbers of graphs, Results Math. 17 (1990) 333-341. Zbl0748.05066
11. [11] L. Volkmann, Foundations of Graph Theory (Springer, Wien, New York, 1996) (in German). Zbl0844.05001
12. [12] L. Volkmann, Some remarks on lower bounds on the p-domination number in trees, J. Combin. Math. Combin. Comput., to appear. Zbl1137.05055
13. [13] B. Xu, E.J. Cockayne, T.W. Haynes, S.T. Hedetniemi and S. Zhou, Extremal graphs for inequalities involving domination parameters, Discrete Math. 216 (2000) 1-10, doi: 10.1016/S0012-365X(99)00251-4. Zbl0954.05037

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