## Connected Global Offensive k-alliances in Graphs

 Lutz Volkmann Lehrstuhl II für Mathematik RWTH Aachen University Templergraben 55, D--52056 Aachen, Germany

## Abstract

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 γok,c(G) is the minimum cardinality of a connected global offensive k-alliance in G.

In this paper we characterize connected graphs G with γok,c(G) = n(G). In the case that δ(G) ≥ k ≥ 2, we also characterize the family of connected graphs G with γok,c(G) = n(G) −1. Furthermore, we present different tight bounds of γok,c(G).

Keywords: alliances in graphs, connected global offensive k-alliance, global offensive k-alliance, domination

2010 Mathematics Subject Classification: 05C69.

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