Geodesic
k-independence stable graphs upon edge removal
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 265-274

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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.
Keywords: k-independence stable graphs, k-independence 
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     title = {k-independence stable graphs upon edge removal},
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Chellali, Mustapha; Haynes, Teresa; Volkmann, Lutz. k-independence stable graphs upon edge removal. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 265-274. http://geodesic.mathdoc.fr/item/DMGT_2010_30_2_a7/