Bounds on the global offensive k-alliance number in graphs

Chellali, Mustapha; Haynes, Teresa W.; Randerath, Bert; Volkmann, Lutz

doi:10.7151/dmgt.1467

Cited by 6 publications

(4 citation statements)

References 8 publications

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“…⌉ (note that these two bounds are attained, for example, for the complete graph K n where γ o k (G) = 1 and k = 3−n). Moreover, Chellali et al [63] obtained different bounds for γ o k (G) in terms of order, maximum degree, independence number, chromatic number and minimum degree. For instance, they proved that if G is a graph of order n with minimum degree δ n ≥ k ≥ 1 (resp.…”

Section: General Graphsmentioning

confidence: 99%

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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Section: General Graphsmentioning

confidence: 99%

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

Ouazine

Slimani

Tari

2018

AKCE International Journal of Graphs and Combinatorics

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“…Combinatorial results on signed total dominating functions can be found in [10,32,33,71]. For combinatorial results on defensive (offensive) kalliances we cite, for instance, [8,25,54,53,55,58,68]. Likewise, complexity results have been obtained in this setting; we only refer to [42] and the literature quoted therein for signed total domination and to [43] for signed total k-domination.…”

Section: Three Variants Of Signed Dominationmentioning

confidence: 99%

A survey on alliances and related parameters in graphs

Fernau

Rodríguez-Velázquez

2014

ejgta

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In this paper, we show that several graph parameters are known in different areas under completely different names. More specifically, our observations connect signed domination, monopolies, α-domination, α-independence, positive influence domination, and a parameter associated to fast information propagation in networks to parameters related to various notions of global r-alliances in graphs. We also propose a new framework, called (global) (D, O)-alliances, not only in order to characterize various known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination. Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations.

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“…The global offensive k-alliance number γ k o (G) is the minimum cardinality of a global offensive k-alliance in G. A global offensive k-alliance set of the minimum cardinality of a graph G is called a γ k o (G)-set. Results on global offensive k-alliances were given, for example, by Bermudo, Rodríguez-Velázquez, Sigarreta and Yero [1], Chellali [2], Chellali, Haynes, Randerath and Volkmann [3] and Fernau, Rodríguez and Sigarreta [4].…”

Section: Terminology and Introductionmentioning

confidence: 99%

Connected global offensive k-alliances in graphs

Volkmann¹

2011

Discuss. Math. Graph Theory

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We consider finite graphs G with vertex set V (G). For a subset S ⊆ V (G), we define by G[S] the subgraph induced by S. By n(G) = |V (G)| and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V (G) is a connected global offensive k-alliance of the connected graph G, if G[S] is connected and |N (v)∩S| ≥ |N (v)−S|+k for every vertex v ∈ V (G)−S, where N (v) is the neighborhood of v. The connected global offensive k-alliance number γ k,c o (G) is the minimum cardinality of a connected global offensive k-alliance in G. In this paper we characterize connected graphs G with γ k,c o (G) = n(G). In the case that δ(G) ≥ k ≥ 2, we also characterize the family of connected graphs G with γ k,c o (G) = n(G)− 1. Furthermore, we present different tight bounds of γ k,c o (G).

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Bounds on the global offensive k-alliance number in graphs

Abstract: Let G = (V (G), E(G)) be a graph, and let k ≥ 1 be an integer.is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on γ k o (G) in terms of order, maximum degree, independence number, chromatic number and minimum degree.

Cited by 6 publications

References 8 publications

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

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

A survey on alliances and related parameters in graphs

Connected global offensive k-alliances in graphs

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