ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 32(2) (2012) 263-270
DOI: 10.7151/dmgt.1603

Trees with Equal 2-domination and 2-independence Numbers

Mustapha Chellali and Nacéra Meddah

LAMDA-RO Laboratory, Department of Mathematics
University of Blida
B.P. 270, Blida, Algeria


Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V −S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ2(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). Fink and Jacobson proved that γ2(G) ≤ β2(G) for every graph G. In this paper we provide a constructive characterization of trees with equal 2-domination and 2-independence numbers.

Keywords: 2-domination number, 2-independence number, trees

2010 Mathematics Subject Classification: 05C69.


[1]M. Borowiecki, On a minimaximal kernel of trees, Discuss. Math. 1 (1975) 3--6.
[2]M. Chellali, O. Favaron, A. Hansberg and L. Volkmann, k-domination and k-independence in graphs: A Survey, Graphs and Combinatorics, 28 (2012) 1--55, doi: 10.1007/s00373-011-1040-3.
[3]O. Favaron, On a conjecture of Fink and Jacobson concerning k-domination and k-dependence, J. Combinat. Theory (B) 39 (1985) 101--102, doi: 10.1016/0095-8956(85)90040-1.
[4]J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science., ed(s), Y. Alavi and A.J. Schwenk (Wiley, New York, 1985) 283--300.
[5]T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs ( Marcel Dekker, New York, 1998).
[6]T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, New York 1998).

Received 14 September 2010
Revised 10 May 2011
Accepted 11 May 2011