Geodesic
Connected global offensive k-alliances in graphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 699-707.

Voir la notice de l'article provenant de la source Library of Science

We consider finite graphs G with vertex set V(G). For a subset S ⊆ V(G), we define by G[S] the subgraph induced by S. By n(G) = |V(G) | and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V(G) is a connected global offensive k-alliance of the connected graph G, if G[S] is connected and |N(v) ∩ S | ≥ |N(v) -S | + k for every vertex v ∈ V(G) -S, where N(v) is the neighborhood of v. The connected global offensive k-alliance number γₒ^k,c(G) is the minimum cardinality of a connected global offensive k-alliance in G.
Keywords: alliances in graphs, connected global offensive k-alliance, global offensive k-alliance, domination 
@article{DMGT_2011_31_4_a5,
     author = {Volkmann, Lutz},
     title = {Connected global offensive k-alliances in graphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {699--707},
     publisher = {mathdoc},
     volume = {31},
     number = {4},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a5/}
}
TY  - JOUR
AU  - Volkmann, Lutz
TI  - Connected global offensive k-alliances in graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2011
SP  - 699
EP  - 707
VL  - 31
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a5/
LA  - en
ID  - DMGT_2011_31_4_a5
ER  - 
%0 Journal Article
%A Volkmann, Lutz
%T Connected global offensive k-alliances in graphs
%J Discussiones Mathematicae. Graph Theory
%D 2011
%P 699-707
%V 31
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a5/
%G en
%F DMGT_2011_31_4_a5
Volkmann, Lutz. Connected global offensive k-alliances in graphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 699-707. http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a5/

[1] S. Bermudo, J.A. Rodriguez-Velázquez, J.M. Sigarreta and I.G. Yero, On global offensive k-alliances in graphs, Appl. Math. Lett. 23 (2010) 1454-1458, doi: 10.1016/j.aml.2010.08.008.

[2] M. Chellali, Trees with equal global offensive k-alliance and k-domination numbers, Opuscula Math. 30 (2010) 249-254.

[3] M. Chellali, T.W. Haynes, B. Randerath and L. Volkmann, Bounds on the global offensive k-alliance number in graphs, Discuss. Math. Graph Theory 29 (2009) 597-613, doi: 10.7151/dmgt.1467.

[4] H. Fernau, J.A. Rodriguez and J.M. Sigarreta, Offensive r-alliance in graphs, Discrete Appl. Math. 157 (2009) 177-182, doi: 10.1016/j.dam.2008.06.001.

[5] J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science (John Wiley and Sons, New York, 1985) 283-300.

[6] J.F. Fink and M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science (John Wiley and Sons, New York, 1985) 301-311.

[7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).

[8] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Domination in Graphs: Advanced Topics ( Marcel Dekker, New York, 1998).

[9] P. Kristiansen, S.M. Hedetniemi and S.T. Hedetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004) 157-177.

[10] O. Ore, Theory of graphs (Amer. Math. Soc. Colloq. Publ. 38 Amer. Math. Soc., Providence, R1, 1962).

[11] K.H. Shafique and R.D. Dutton, Maximum alliance-free and minimum alliance-cover sets, Congr. Numer. 162 (2003) 139-146.

[12] K.H. Shafique and R.D. 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.