Discussiones Mathematicae Graph Theory 28(1) (2008) 151-163
doi: 10.7151/dmgt.1398

[BIBTex] [PDF] [PS]

MAXIMAL k-INDEPENDENT SETS IN GRAPHS

1Mostafa Blidia, 1Mustapha Chellali, 2Odile Favaron  and  1Nacéra Meddah

1LAMDA-RO Laboratory, Department of Mathematics
University of Blida
B.P. 270, Blida, Algeria
e-mail: mblidia@hotmail.com, m_chellali@yahoo.com

2Univ. Paris-Sud
LRI, URM 8623, Orsay, F-91405, France
CNRS, Orsay, F91405
e-mail: of@lri.fr

Abstract

A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted ik(G) and βk(G). We give some relations between βk(G) and βj(G) and between ik(G) and ij(G) for j ≠ k. We study two families of extremal graphs for the inequality i2(G) ≤ i(G)+β(G). Finally we give an upper bound on i2(G) and a lower bound when G is a cactus.

Keywords: k-independent, cactus.

2000 Mathematics Subject Classification: 05C69.

References

[1] M. Blidia, M. Chellali, O. Favaron and N. Meddah, On k-independence in graphs with emphasis on trees, Discrete Math. 307 (2007) 2209-2216, doi: 10.1016/j.disc.2006.11.007.
[2] M. Borowiecki and D. Michalak, Generalized independence and domination in graphs, Discrete Math. 191 (1998) 51-56, doi: 10.1016/S0012-365X(98)00092-2.
[3] O. Favaron, On a conjecture of Fink and Jacobson concerning k-domination and k-dependence, J. Combin. Theory (B) 39 (1985) 101-102, doi: 10.1016/0095-8956(85)90040-1.
[4] O. Favaron, k-domination and k-independence in graphs, Ars Combin. 25 C (1988) 159-167.
[5] J.F. Fink and M.S. Jacobson, n-domination, n-dependence and forbidden subgraphs, Graph Theory with Applications to Algorithms and Computer (John Wiley and sons, New York, 1985) 301-311.
[6] G. Chartrand and L. Lesniak, Graphs & Digraphs: Third Edition (Chapman & Hall, London, 1996).
[7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).

Received 18 January 2007
Revised 8 January 2008
Accepted 8 January 2008