Small alliances in a weighted graph

Small alliances in a weighted graph

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 pow...

Small Alliances in Graphs

Let G = (V, E) be a graph. A nonempty subset S ⊆ V is a (strong defensive) alliance of G if every node in S has at least as many neighbors in S than in V \S. This work is motivated by the following observation: when G is a locally structured graph its nodes typically belong to small alliances. Despite the fact that finding the smallest alliance in a graph is NP-hard, we can at least compute in ...

Partitioning a graph into offensive k-alliances

A defensive k-alliance in a graph is a set S of vertices with the property that every vertex in S has at least k more neighbors in S than it has outside of S. A defensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive k-alliances. The (global) defensive k-alliance partition number o...

A Novel Molecular Descriptor Derived from Weighted Line Graph

The Bertz indices, derived by counting the number of connecting edges of line graphs of a molecule were used in deriving the QSPR models for the physicochemical properties of alkanes. The inability of these indices to identify the hetero centre in a chemical compound restricted their applications to hydrocarbons only. In the present work, a novel molecular descriptor has been derived from the w...