Weighted Alliances in Graphs

Let G = (V, E) be a graph and let W:V→N be a non-negative integer weighting of the vertices in V. A nonempty set of vertices S ⊆ V is called a weighted defensive alliance if ∀v ∈ S ,∑u∈N[v]∩S w(u) ≥ ∑x∈N(v)−S w(x). A non-empty set S ⊆ V is a weighted offensive alliance if ∀v ∈ δS ,∑u∈N(v)∩S w(u) ≥ ∑x∈N[v]−S w(x). A weighted alliance which is both defensive and offensive is called a weighted powerful alliance, (awp(G) denotes the weighted powerful alliance number of a graph G) while a weighted alliance S for which δS = V −S is called a weighted global alliance. This paper shows that the algorithmic complexity of finding the weighted alliance number for all types of alliances is NP-complete even when restricted to stars, while algorithms are presented for finding weighted defensive and weighted offensive alliances on paths.