Partitioning a Graph into Global Powerful k-Alliances

Yero, Ismael G.; Rodríguez-Velázquez, Juan A.

doi:10.1007/s00373-011-1065-7

Cited by 5 publications

(13 citation statements)

References 12 publications

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“…the difference between the number of outside and inside neighbours should be larger than a given constant, the community should also be a dominating set, see e.g. [4,5,15] for overview and further references.…”

Section: Introductionmentioning

confidence: 99%

Structural and Algorithmic Properties of 2-Community Structures

2017

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We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331-336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V | − 3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.

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Section: Introductionmentioning

confidence: 99%

Structural and Algorithmic Properties of 2-Community Structures

2017

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“…On the other hand, there exist several studies on partitioning graphs into k-alliances and various theoretical results are obtained for some graphs classes in the literature. In this framework, the partitioning of graphs into defensive k-alliances is investigated by Eroh and Gera [20,21], Haynes and Lachniet [22], Yero et al [23], the partitioning into offensive k-alliances is the subject matter of Sigarreta et al [24], Yero [25], and the partitioning into powerful k-alliances is discussed in Slimani and Kheddouci [26], Yero and Rodríguez-Velázquez [27]. In this context, it would be interesting to see how such partitioning can be of interest to solve practical problems.…”

Section: Historical View and Applications Of Alliancesmentioning

confidence: 99%

“…Cartesian product graphs Let G i = (V i , E i ) be a graph of order n i , minimum degreeδ i and maximum degree∆ i , i ∈ {1, 2}. Yero [25] and Yero and Rodríguez-Velázquez [27] studied the powerful k-alliances in Cartesian product graphs and obtained some results for the associated parameters. Thus, they showed that if…”

Section: Path Graphsmentioning

confidence: 99%

“…• [25,27] • γ p k (G 1 × G 2 ) ≤ γ p k 1 (G 1 )n 2 [25,27] upper powerful (−1)-alliance number A p −1 (G), the upper powerful 0-alliance number A p 0 (G) and the upper powerful k-alliance number A p k (G) are not studied for all graph classes (note that these three latter numbers which are not defined in the literature can be similarly defined as in the cases of defensive and offensive k-alliances). Also, the classes of bipartite graphs and line graphs are not studied for this kind of alliances.…”

Section: Graph Classesmentioning

confidence: 99%

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

Ouazine

Slimani

Tari

2018

AKCE International Journal of Graphs and Combinatorics

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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

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“…Since then many variatons appeared. The most extensively studied are defensive alliances [9,8,13,19,21], offensive alliances [7,12,18] and powerful or dual alliances [2,3,22]. A more generalized concept of alliance is represented by k-alliances [1,15,16,17,19], and Dourado et al presented a new definition of alliances, namely, (f, g)-alliances [6], that generalizes previous concepts.…”

Section: Introductionmentioning

confidence: 99%