PDF

Discussiones Mathematicae Graph Theory 27(1) (2007) 93-103
DOI: https://doi.org/10.7151/dmgt.1347

CHARACTERIZATION OF BLOCK GRAPHS WITH EQUAL 2-DOMINATION NUMBER AND DOMINATION NUMBER PLUS ONE

Adriana Hansberg  and  Lutz Volkmann

Lehrstuhl II für Mathematik
RWTH Aachen University
52056 Aachen, Germany
e-mail: hansberg@math2.rwth-aachen.de
e-mail: volkm@math2.rwth-aachen.de

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 γ_p(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.

Keywords: domination, 2-domination, multiple domination, block graph.

2000 Mathematics Subject Classification: 05C69.

References

[1]	M. Blidia, M. Chellali and L. Volkmann, Bounds of the 2-domination number of graphs, Utilitas Math. 71 (2006) 209-216.
[2]	J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science (John Wiley and Sons, New York, 1985), 282-300.
[3]	J.F. Fink and M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science (John Wiley and Sons, New York, 1985), 301-311.
[4]	J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079.
[5]	T. Gallai, Über extreme Punkt-und Kantenmengen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 2 (1959) 133-138.
[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]	C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104.
[9]	B. Randerath and L. Volkmann, Characterization of graphs with equal domination and covering number, Discrete Math. 191 (1998) 159-169, doi: 10.1016/S0012-365X(98)00103-4.
[10]	J. Topp and L. Volkmann, On domination and independence numbers of graphs, Results Math. 17 (1990) 333-341.
[11]	L. Volkmann, Foundations of Graph Theory (Springer, Wien, New York, 1996) (in German).
[12]	L. Volkmann, Some remarks on lower bounds on the p-domination number in trees, J. Combin. Math. Combin. Comput., to appear.
[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.

Received 7 December 2005
Revised 18 October 2006