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## Opuscula Mathematica

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### Trees with equal global offensive k-alliance and k-domination numbers

Autorzy Chellali, M.
Treść / Zawartość
Warianty tytułu
Języki publikacji EN
Abstrakty
 EN Let k ≥ 1 be an integer. A set S of vertices of a graph G = (V (G), E(G)) is called a global offensive k-alliance if |N(v) ∩ S| ≥ |N(v) - S| + k for every v ∈ V (G) - S, where N(v) is the neighborhood of v. The subset S is a k-dominating set of G if every vertex in V (G) - S has at least k neighbors in S. The global offensive k-alliance number [formula] is the minimum cardinality of a global offensive k-alliance in G and the k-domination number ϒ k(G) is the minimum cardinality of a k-dominating set of G. For every integer k ≥ 1 every graph G satisfies [formula]. In this paper we provide for k ≥ 2 a characterization of trees T with equal [formula] and ϒ k(T).
Słowa kluczowe
 EN global offensive k-alliance number   k-domination number   trees
Wydawca AGH University of Science and Technology Press
Czasopismo Opuscula Mathematica
Rocznik 2010
Tom Vol. 30, no. 3
Strony 249--254
Opis fizyczny Bibliogr. 11 poz.
Twórcy
 autor Chellali, M. University of Blida LAMDA-RO Laboratory, Department of Mathematics B.P. 270, Blida, Algeria, m_chellali@yahoo.com
Bibliografia
[1] M. Blidia, M. Chellali, L. Volkmann, Some bounds on the p-domination number in trees, Discrete Math. 306 (2006), 2031–2037.
[2] M. Bouzefrane, M. Chellali, On the global offensive alliance number of a tree, Opuscula Math. 29 (2009), 223–228.
[3] M. Chellali, T.W. Haynes, B. Randerath, L. Volkmann, Bounds on the global offensive k-alliance number in graphs, Discuss. Math. Graph Theory 29 (2009), 597–613.
[4] O. Favaron, A. Hansberg, L. Volkmann, On k-domination and minimum degree in graphs, J. Graph Theory 57 (2008), 33–40.
[5] H. Fernau, J.A. Rodríguez, J.M. Sigarreta, Offensive r-alliance in graphs, Discrete Appl. Math. 157 (2009), 177–182.
[6] J.F. Fink, M.S. Jacobson, n-domination in graphs, Graph Theory with Applications to Algorithms and Computer Science, John Wiley and Sons, New York, 1985, 282–300.
[7] J.F. Fink, M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, Graph Theory with Applications to Algorithms and Computer Science, John Wiley and Sons, New York 1985, 301–311.
[8] A. Hansberg, L. Volkmann, Lower bounds on the p-domination number in terms of cycles and matching number, J. Combin. Math. Combin. Comput. 68 (2009), 245–255.
[9] P. Kristiansen, S.M. Hedetniemi, S.T. Hedetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177.
[10] K.H. Shafique, R.D. Dutton, Maximum alliance-free and minimum alliance-cover sets, Congr. Numer. 162 (2003), 139–146.
[11] K.H. Shafique, R. Dutton, A tight bound on the cardinalities of maximum alliance-free and minimum alliance-cover sets, J. Combin. Math. Combin. Comput. 56 (2006), 139–145.
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