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

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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


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.


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