Connected global offensive k-alliances in graphs

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 γ k,c o (G) is the minimum cardinality of a connected global offensive k-alliance in G. In this paper we characterize connected graphs G with γ k,c o (G) = n(G). In the case that δ(G) ≥ k ≥ 2, we also characterize the family of connected graphs G with γ k,c o (G) = n(G)− 1. Furthermore, we present different tight bounds of γ k,c o (G).

Alliances in graphs: Parameters, properties and applications—A survey

“…Also they showed that if G is a graph of order n, minimum degree δ n and maximum degree δ 1 , then γ o k (G) ≥ n(δ n +k) 2δ 1 +δ n +k . Besides, Volkmann [64] investigated the connected global offensive k-alliance number γ o k,c (G) which is the minimum cardinality of a connected global offensive k-alliance in the graph G. In this framework, he characterized connected graphs G with γ o k,c (G) = n(G) (γ o k,c (G) = n(G) − 1 in the case that δ n ≥ k ≥ 2), and presented different tight bounds for γ o k,c (G). Yero and Rodríguez-Velázquez [32] studied the mathematical properties of boundary powerful k-alliances and obtained that if S is a boundary offensive k-alliance in a graph G, then…”

A survey on alliances and related parameters in graphs

“…• Again in domination theory, dominating sets with additional properties like connectedness or independence have been thoroughly studied. In the context of alliances, we are only aware of one such paper, namely [64]. In particular those connected alliances should make very much sense in view of the original motivation of alliances.…”