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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1347,
author = {Adriana Hansberg and Lutz Volkmann},
title = {Characterization of block graphs with equal 2-domination number and domination number plus one},
journal = {Discussiones Mathematicae Graph Theory},
volume = {27},
year = {2007},
pages = {93-103},
zbl = {1189.05127},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1347}
}
Adriana Hansberg; Lutz Volkmann. Characterization of block graphs with equal 2-domination number and domination number plus one. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 93-103. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1347/
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