Let G = (V,E) be a graph. The distance between two vertices u and v in a connected graph G is the length of the shortest (u-v) path in G. A set D ⊆ V(G) is a dominating set if every vertex of G is at distance at most 1 from an element of D. The domination number of G is the minimum cardinality of a dominating set of G. A set D ⊆ V(G) is a 2-distance dominating set if every vertex of G is at distance at most 2 from an element of D. The 2-distance domination number of G is the minimum cardinality of a 2-distance dominating set of G. We characterize all trees and all unicyclic graphs with equal domination and 2-distance domination numbers.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1552,
author = {Joanna Raczek},
title = {Graphs with equal domination and 2-distance domination numbers},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {375-385},
zbl = {1234.05179},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1552}
}
Joanna Raczek. Graphs with equal domination and 2-distance domination numbers. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 375-385. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1552/
[000] [1] M. Borowiecki and M. Kuzak, On the k-stable and k-dominating sets of graphs, in: Graphs, Hypergraphs and Block Systems. Proc. Symp. Zielona Góra 1976, ed. by M. Borowiecki, Z. Skupień, L. Szamkołowicz, (Zielona Góra, 1976). | Zbl 0344.05143
[001] [2] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker Inc., 1998). | Zbl 0890.05002