# k-independence stable graphs upon edge removal

• Volume: 30, Issue: 2, page 265-274
• ISSN: 2083-5892

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## Abstract

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Let k be a positive integer and G = (V(G),E(G)) a graph. A subset S of V(G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k-1. The maximum cardinality of a k-independent set of G is the k-independence number βₖ(G). A graph G is called β¯ₖ-stable if βₖ(G-e) = βₖ(G) for every edge e of E(G). First we give a necessary and sufficient condition for β¯ₖ-stable graphs. Then we establish four equivalent conditions for β¯ₖ-stable trees.

## How to cite

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Mustapha Chellali, Teresa W. Haynes, and Lutz Volkmann. "k-independence stable graphs upon edge removal." Discussiones Mathematicae Graph Theory 30.2 (2010): 265-274. <http://eudml.org/doc/270891>.

@article{MustaphaChellali2010,
abstract = {Let k be a positive integer and G = (V(G),E(G)) a graph. A subset S of V(G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k-1. The maximum cardinality of a k-independent set of G is the k-independence number βₖ(G). A graph G is called β¯ₖ-stable if βₖ(G-e) = βₖ(G) for every edge e of E(G). First we give a necessary and sufficient condition for β¯ₖ-stable graphs. Then we establish four equivalent conditions for β¯ₖ-stable trees.},
author = {Mustapha Chellali, Teresa W. Haynes, Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {k-independence stable graphs; k-independence; -independence stable graphs; -independence},
language = {eng},
number = {2},
pages = {265-274},
title = {k-independence stable graphs upon edge removal},
url = {http://eudml.org/doc/270891},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Mustapha Chellali
AU - Teresa W. Haynes
AU - Lutz Volkmann
TI - k-independence stable graphs upon edge removal
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 2
SP - 265
EP - 274
AB - Let k be a positive integer and G = (V(G),E(G)) a graph. A subset S of V(G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k-1. The maximum cardinality of a k-independent set of G is the k-independence number βₖ(G). A graph G is called β¯ₖ-stable if βₖ(G-e) = βₖ(G) for every edge e of E(G). First we give a necessary and sufficient condition for β¯ₖ-stable graphs. Then we establish four equivalent conditions for β¯ₖ-stable trees.
LA - eng
KW - k-independence stable graphs; k-independence; -independence stable graphs; -independence
UR - http://eudml.org/doc/270891
ER -

## References

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1. [1] M. Blidia, M. Chellali and L. Volkmann, Some bounds on the p-domination number in trees, Discrete Math. 306 (2006) 2031-2037, doi: 10.1016/j.disc.2006.04.010. Zbl1100.05069
2. [2] J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer (John Wiley and sons, New York, 1985) 283-300.
3. [3] G. Gunther, B. Hartnell and D.F. Rall, Graphs whose vertex independence number is unaffected by single edge addition or deletion, Discrete Appl. Math. 46 (1993) 167-172, doi: 10.1016/0166-218X(93)90026-K. Zbl0792.05116

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