Given a simple graph G = (V,E), a subset S of the vertices is called a global defensive alliance if S is a dominating set and for every vertex v in S at least half of the vertices in the closed neighborhood of v are in S. Similarly, a subset S is called a global offensive alliance if S is a dominating set and for every vertex v not in S at least half of the vertices in the closed neighborhood o...

A global defensive (respectively, offensive) alliance in a graph G = (V,E) is a set of vertices S ⊆ V with the properties that every vertex in V − S has at least one neighbor in S, and for each vertex v in S (respectively, in V − S) at least half the vertices from the closed neighborhood of v are in S. These alliances are called strong if a strict majority of vertices from the closed neighborho...

The concepts of secure sets and security number in graphs were introduced first by Brigham et al. in 2007 as a generalization of the concept of alliances in graphs. Defensive alliances are related to the defense of a single vertex. But, in a general realistic settings, alliances should be formed so that any attack on the entire alliance or any subset of the alliance can be defended. In this sen...

This essay explores the distinction between federations and alliances and asks the question: When will states choose to federate rather than ally? William Riker (1964) argues that a necessary condition for a federal state’s formation is that those offering the federal bargain must seek to "expand their territorial control, usually either to meet an external military or diplomatic threat or to p...

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 (where k is the strength of defensive k−alliance). An offensive k−alliance 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 th...

We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) that allows the formation of two defensive alliances consisting of the even and odd label species. The species are distributed on the sites of a square lattice. The evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly: if the sit...

We study a set of six-species ecological models where each species has two predators and two prey. On a square lattice the time evolution is governed by iterated invasions between the neighboring predator-prey pairs chosen at random and by a site exchange with a probability Xs between the neutral pairs. These models involve the possibility of spontaneous formation of different defensive allianc...

A global boundary defensive k-alliance in a graph G = (V,E) is a dominating set S of vertices of G with the property that every vertex in S has k more neighbors in S than it has outside of S. A global boundary offensive k-alliance in a graph G is a set S of vertices of G with the property that every vertex in V − S has k more neighbors in S than it has outside of S. We define a global boundary ...

the United States’ desire for isolationism and its desire to maintain hegemony. While the United States has initiated and continued wars in Afghanistan and Iraq that are increasingly unpopular with alliance members, it has also focused on increasing NATO buy-in and has become involved in ance members. This raises the question of whether strong states suffer from alliance entrapment, a that are ...