Let Γ(V,E) be a simple connected graph with more than one vertex, without loops or multiple edges. A nonempty subset S⊆V is a global offensive alliance if every vertex v∈V−S satisfies that δS(v)≥δS¯(v)+1. The global offensive alliance numberγo(Γ) is defined as the minimum cardinality among all global offensive alliances. Let R be a finite commutative ring with identity. In this paper, we study the global offensive alliance number of the zero-divisor graph Γ(R).

<abstract><p>A nonempty subset $ D $ of vertices in a graph $ \Gamma = (V, E) $ is said is an <italic>offensive alliance</italic>, if every vertex $ v \in \partial(D) $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum offensive alliance of $ \Gamma $ is called the <italic>offensive alliance number</italic> $ \alpha ^o(\Gamma) $ of $ \Gamma $. An offensive alliance $ D $ is called <italic>global</italic>, if every $ v \in V - D $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum global offensive alliance of $ \Gamma $ is called the <italic>global offensive alliance number</italic> $ \gamma^o(\Gamma) $ of $ \Gamma $. For a finite commutative ring with identity $ R $, $ \Gamma(R) $ denotes the zero divisor graph of $ R $. In this paper, we compute the offensive alliance (global, independent, and independent global) numbers of $ \Gamma(\mathbb{Z}_n) $, for some cases of $ n $.</p></abstract>

Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace C(G) of a dendrite G; then, we give necessary and sufficient conditions so that the aforementioned result can be applied when G is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space X for which the anchored hyperspace Cp(X) is an arc for some p∈X.

Let Γ(V, E) be a simple graph without loops nor multiple edges. A nonempty subset S ⊆ V is said a global offensive alliance if every vertex v ∈ V − S satisfies that δ S (v) ≥ δ S (v) + 1. The global offensive alliance number γ o (Γ) is defined as the minimum cardinality among all global offensive alliances. Let R be a finite commutative ring with identity. In this paper, we initiate the study of the global offensive alliance number of the zero-divisor graph Γ(R).