# Maximal k-independent sets in graphs

Mostafa Blidia; Mustapha Chellali; Odile Favaron; Nacéra Meddah

Discussiones Mathematicae Graph Theory (2008)

- Volume: 28, Issue: 1, page 151-163
- ISSN: 2083-5892

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topMostafa Blidia, et al. "Maximal k-independent sets in graphs." Discussiones Mathematicae Graph Theory 28.1 (2008): 151-163. <http://eudml.org/doc/270197>.

@article{MostafaBlidia2008,

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 $β_j(G)$ and between iₖ(G) and $i_j(G)$ 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.},

author = {Mostafa Blidia, Mustapha Chellali, Odile Favaron, Nacéra Meddah},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {k-independent; cactus; -independent},

language = {eng},

number = {1},

pages = {151-163},

title = {Maximal k-independent sets in graphs},

url = {http://eudml.org/doc/270197},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Mostafa Blidia

AU - Mustapha Chellali

AU - Odile Favaron

AU - Nacéra Meddah

TI - Maximal k-independent sets in graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2008

VL - 28

IS - 1

SP - 151

EP - 163

AB - 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 $β_j(G)$ and between iₖ(G) and $i_j(G)$ 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.

LA - eng

KW - k-independent; cactus; -independent

UR - http://eudml.org/doc/270197

ER -

## References

top- [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. Zbl1123.05066
- [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. Zbl0958.05102
- [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. Zbl0583.05049
- [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). Zbl0890.05002

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