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Discussiones Mathematicae Graph Theory 30(2) (2010) 265-274
DOI: https://doi.org/10.7151/dmgt.1492

k-INDEPENDENCE STABLE GRAPHS UPON EDGE REMOVAL

Mustapha Chellali LAMDA-RO Laboratory, Department of Mathematics University of Blida B.P. 270, Blida, Algeria e-mail: [email protected]

Teresa W. Haynes Department of Mathematics, East Tennessee State University Johnson City, TN 37614 USA e-mail: [email protected]

Lutz Volkmann Lehrstuhl II für Mathematik, RWTH Aachen University Templergraben 55, D-52056 Aachen, Germany e-mail: [email protected]

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 β_k(G). A graph G is called β_k^--stable if β_k(G-e) = β_k(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.

2010 Mathematics Subject Classification: 05C69.

References

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[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]	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.

Received 6 December 2008
Revised 30 June 2009
Accepted 30 June 2009