A defensive k-alliance in a graph G = (V, E) is a set of vertices A ⊆ V such that for every vertex v ∈ A, the number of neighbors v has in A is at least k more than the number of neighbors it has in V − A (k is a measure of the strength of alliance). In this paper, we deal with two types of sets associated with defensive k-alliances; maximum defensive k-alliance free and minimum defensive k-all...

Let Γ be a simple graph of size m and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn. Let L(Γ) denotes the line graph of Γ. The aim of this paper is to study mathematical properties of the alliance number, a(L(Γ), and the global alliance number, γa(L(Γ)), of the line graph of a simple graph. We show that ⌈

Let Γ = (V,E) be a simple graph. For a nonempty set X ⊆ V , and a vertex v ∈ V , δX(v) denotes the number of neighbors v has in X. A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E) if δS(v) ≥ δS̄(v)+k, ∀v ∈ S. A defensive k-alliance S is called global if it forms a dominating set. The global defensive k-alliance number of Γ, denoted by γ k(Γ), is the minimum cardinality of a defensive ...

A defensive (offensive) k-alliance in Γ = (V,E) is a set S ⊆ V such that for every v ∈ S (v ∈ ∂S), the number of neighbors v has in S is at least k more than the number of neighbors it has in V \ S. A set X ⊆ V is defensive (offensive) k-alliance free, if for all defensive (offensive) k-alliance S, S \ X 6= ∅, i.e., X do not contain any defensive (offensive) k-alliance as a subset. A set Y ⊆ V ...

A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbors is in S. The problem of finding a defensive alliance of minimum size in a given graph is NP-hard and there are polynomial-time algorithms if certain parameters are bounded by a fixed constant. In particular, fixed-parameter tractability results have been obtained for some structural p...

States form defensive military alliances to enhance their security in the face of potential or realized interstate conflict. The network of these international alliances is increasingly interconnected, now linking most of the states in a complex web of ties. These alliances can be used both as a tool for securing cooperation and to foster peace between direct partners. However, do indirect conn...

Simple combinations of common competitive mechanisms can easily result in cyclic competitive dominance relationships between species. The topological features of such competitive networks allow for complex spatial coexistence patterns. We investigate self-organization and coexistence in a lattice model, describing the spatial population dynamics of competing bacterial strains. With increasing d...