Alliances in graphs: Parameters, properties and applications—A survey. Issue 2 (1st August 2018) - Ouazine, Kahina; Slimani, Hachem; Tari, Abdelkamel
Abstract: In practice, an alliance can be a bond or connection between individuals, families, states, or entities, etc. Formally, a non-empty set S of vertices of a graph G is a defensive k -alliance (resp. an offensive k -alliance) if every vertex of S (resp. the boundary of S ) has at least k more neighbors inside of S than it has outside of S . A powerful k -alliance is both defensive k -alliance and offensive ( k + 2 ) -alliance. During the last decade there has been a remarkable development on these three kinds of alliances. Due to their variety of applications, the alliances in its broad sense have received a special attention from many scientists and researchers. There have been applications of alliances in several areas such as bioinformatics, distributed computing, web communities, social networks, data clustering, business, etc. Several k -alliance numbers have been defined and a huge number of theoretical (algorithmic and computational) results are obtained for various graph classes. In this paper, we present a survey which covers a number of practical applications of alliances and the vast mathematical properties of the three types of k -alliances by giving a special attention to the study of the associated k -alliance (partition) numbers for different graph classes.