Alliances in graphs: Parameters, properties and applications—A survey

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. c

Solutions to four open problems on quorum colorings of graphs

“…Quorum colorings have several real-world applications (cf. [7,9] and [12]), including data clustering, the goal of which is to partition a dataset into homogeneous packets in the sense that the data in the same packet share more characteristics in common between them than with data outside of this packet. This problem can be modeled by a graph G in which each data is represented by a vertex so that two vertices are adjacent if the corresponding data share a fixed minimum number of common characteristics, and hence the objective is to color the vertex set of the resulting graph such that at least half of the neighbors of each vertex v have the same color as v, where v is counted itself as a neighbor.…”

Reducing saturation and congestion in VANET networks: Alliance‐based approach and comparisons

“…The research field of alliances as a concept of graph theory has grown considerably due to their interesting applications in various domains, including distributed systems, mobile ad hoc networks (MANETs), social networks, data clustering, bioinformatics, and web communities . In practice, an alliance can be a link or connection between individuals, families, states, or parties.…”

“…In the literature, there are a huge number of studies on the mathematical properties of different types of alliances in graphs. For more details on the properties and the applications of k ‐alliances in practice, the readers can refer to Ouazine et al and Yero and Rodríguez‐Velázquez . Note that an alliance S of any type (defensive, offensive, or powerful) is called global , if S is a dominating set in the graph G ; ie, every vertex outside of S is adjacent to at least one vertex inside of S .…”

Global defensive k-alliances in directed graphs: combinatorial and computational issues

“…Remarkable examples are the articles [14,15], where alliances were generalized to k-alliances. To see more information on alliances in graphs we suggest the recent surveys [2,11,19] and references cited therein.…”

“…The global defensive k-alliance number of G, denoted by γ d k (G), is the minimum cardinality of a global defensive k-alliance in G. Since then a high number of papers on different versions of alliances in graphs have been published in the literature and some interesting real-world applications can be found among them (the reader can refer to [8,12,14] for instance). To have a better overview on all the information about alliances we suggest the fairly complete surveys [2,11,19].…”