Chellali, Mustapha; Haynes, Teresa W.; Volkmann, Lutz \(k\)-independence stable graphs upon edge removal. (English) Zbl 1214.05103 Discuss. Math., Graph Theory 30, No. 2, 265-274 (2010). Summary: 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 \(\beta_k(G)\). A graph \(G\) is called \(\beta^-_k\)-stable if \(\beta_k(G- e)= \beta_k(G)\) for every edge \(e\) of \(E(G)\). First we give a necessary and sufficient condition for \(\beta^-_k\)-stable graphs. Then we establish four equivalent conditions for \(\beta^-_k\)-stable trees. Cited in 1 Document MSC: 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) Keywords:\(k\)-independence stable graphs; \(k\)-independence PDFBibTeX XMLCite \textit{M. Chellali} et al., Discuss. Math., Graph Theory 30, No. 2, 265--274 (2010; Zbl 1214.05103) Full Text: DOI