Geodesic
A Characterization of Trees for a New Lower Bound on the K-Independence Number
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 395-410

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Let k be a positive integer and G = (V,E) a graph of order n. A subset S of V is a k-independent set of G if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1. The maximum cardinality of a k-independent set of G is the k-independence number β_k (G). In this paper, we show that for every graph G, β_k (G) ≥ ⌈ ( n + ( χ(G)-1) Σ_v ∈ S(G)min ( | L_v|, k-1) ) / χ(G) ⌉, where χ(G), s(G) and L_v are the chromatic number, the number of supports vertices and the number of leaves neighbors of v, in the graph G, respectively. Moreover, we characterize extremal trees attaining these bounds.
Keywords: domination, independence, k-independence 
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Meddah, Nacéra; Blidia, Mostafa. A Characterization of Trees for a New Lower Bound on the K-Independence Number. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 395-410. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a12/