On the global offensive alliance number of a tree

Abstract

Abstract. For a graph G = (V, E), a set S ⊆ V is a dominating set if every vertex in V − S has at least a neighbor in S. A dominating set S is a global offensive alliance if for every vertex v in V − S, at least half of the vertices in its closed neighborhood are in S. The domination number γ(G) is the minimum cardinality of a dominating set of G and the global offensive alliance number γo(G) is the minimum cardinality of a global offensive alliance of G. We first show that every tree of order at least three with leaves and s support vertices satisfies γo(T ) ≥ (n − + s + 1)/3 and we characterize extremal trees attaining this lower bound. Then we give a constructive characterization of trees with equal domination and global offensive alliance numbers.

On the global offensive alliance in unicycle graphs

“…Observation 1 If G is a connected graph of order at least three, then there is a γ o (G)-set that contains all the support vertices. [1] gave a lower bound of the global offensive alliance of trees and characterized all extremal trees attaining this bound by considering a family F of trees of order at least three that can be obtained from r disjoint stars by first adding r − 1 edges so that they are incident only with centers of the stars and the resulting graph is connected, and then subdividing each new edge exactly once. They prove the following result.…”

“…Offensive alliances in graphs were be studied in [1,2,3]. In this paper we give a lower bound on the global offensive alliance number.…”

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

“…). Moreover, Bouzefrane and Chellali [65] showed that for a tree T of order n ≥ 3 with l leaves and s support vertices, the global offensive 1-alliance number is bounded lowerly by γ o 1 (T ) ≥ n−l+s+1 3 (with equality if and only if T belongs to a special family of trees F [65]). They also proved that if T ∈ F then γ o 1 (T ) = γ (T ).…”

Trees with unique minimum global offensive alliance

“…is called global offensive alliance set of G and abbreviated GOA-set of G. The global offensive alliance number γ o (G) is the minimum cardinality among all GOA-sets of G. A GOA-set of G of cardinality γ o (G) is called γ o -set of G, or γ o (G)-set. Several works have been carried out on global offensive alliances in graphs (see, for example, [2,6], and elsewhere). Graphs with unique minimum µ-set, where µ is a some graph parameter, is another concept to which much attention was given during the last two decades.…”