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

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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 
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     author = {Volkmann, Lutz},
     title = {Connected global offensive k-alliances in graphs},
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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/