Offensive alliances in graphs

Favaron, Odile; Fricke, Gerd H.; Goddard, Wayne; Hedetniemi, Stephen T.; Kristiansen, Petter; Laskar, Renu C.; Skaggs, R. Duane

doi:10.7151/dmgt.1230

Cited by 57 publications

(39 citation statements)

References 3 publications

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“…In these papers they proposed different types of alliances: namely, defensive, offensive and dual or powerful alliances. For instance, a defensive alliance [9,10,12,14,17] of a graph is a set S of vertices of with the property that every vertex in S has at most one more neighbor outside of S than it has in S. An offensive alliance [6,14,16,17,21] of a graph is a set S of vertices of with the property that every vertex in the neighborhood of S has at least one more neighbor in S than it has outside of S. A powerful alliance [2,3,7,24] is a combination of both, defensive and offensive alliances.…”

Section: Introductionmentioning

confidence: 99%

Partitioning a Graph into Global Powerful k-Alliances

Yero

Rodríguez-Velázquez

2011

Graphs and Combinatorics

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A set S of vertices of a graph is a defensive k-alliance if every vertex v ∈ S has at least k more neighbors in S than it has outside of S. Analogously, a set S is an offensive k-alliance if every vertex in the neighborhood of S has at least k more neighbors in S than it has outside of S. Also, a powerful k-alliance is a set S of vertices of the graph, which is both defensive k-alliance and offensive (k + 2)-alliance. A powerful k-alliance is called global if it is a dominating set. In this paper we show that for k ≥ 0, no graph is partitionable into global powerful k-alliances and, for k ≤ −1, we obtain upper bounds on the maximum number of sets belonging to a partition of a graph into global powerful k-alliances. In addition, we study the close relationships that exist between partitions of a Cartesian product graph, 1 × 2 , into (global) powerful (k 1 + k 2 )-alliances and partitions of i into (global) powerful k i -alliances, i ∈ {1, 2}.

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Section: Introductionmentioning

confidence: 99%

Partitioning a Graph into Global Powerful k-Alliances

Yero

Rodríguez-Velázquez

2011

Graphs and Combinatorics

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“…A defensive alliance is a set of vertices where each vertex has a majority of its neighbors in the alliance. The complexity of finding alliances of a given size k is NP-complete [Cami et al 2006;Favaron et al 2002;Jamieson et al 2002;Shafique 2001], but is fixed parameter tractable [Fernau and Raible 2007].…”

Section: Definitions Of Communitiesmentioning

confidence: 99%

Expanding network communities from representative examples

Mehler

Skiena

2009

ACM Trans. Knowl. Discov. Data

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We present an approach to leverage a small subset of a coherent community within a social network into a much larger, more representative sample. Our problem becomes identifying a small conductance subgraph containing many (but not necessarily all) members of the given seed set. Starting with an initial seed set representing a sample of a community, we seek to discover as much of the full community as possible.We present a general method for network community expansion, demonstrating that our methods work well in expanding communities in real world networks starting from small given seed groups (20 to 400 members). Our approach is marked by incremental expansion from the seeds with retrospective analysis to determine the ultimate boundaries of our community. We demonstrate how to increase the robustness of the general approach through bootstrapping multiple random partitions of the input set into seed and evaluation groups.We go beyond statistical comparisons against gold standards to careful subjective evaluations of our expanded communities. This process explains the causes of most disagreement between our expanded communities and our gold-standards-arguing that our expansion methods provide more reliable communities than can be extracted from reference sources/gazetteers such as Wikipedia.

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“…Alliances in graphs were introduced by Hedetniemi, Hedetniemi, and Kristiansen in [5]. For the study of offensive alliances we cite for example [1] and [2]. For a comprehensive treatment of domination in graphs, see the monographs by Haynes, Hedetniemi, and Slater [3,4].…”

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confidence: 99%

On the global offensive alliance number of a tree

Bouzefrane¹,

Chellali²

2009

OpMath

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Abstract. For a graph G = (V, E), a set S ⊆ V is a dominating set if every vertex in V − S has at least a neighbor in S. A dominating set S is a global offensive alliance if for every vertex v in V − S, at least half of the vertices in its closed neighborhood are in S. The domination number γ(G) is the minimum cardinality of a dominating set of G and the global offensive alliance number γo(G) is the minimum cardinality of a global offensive alliance of G. We first show that every tree of order at least three with leaves and s support vertices satisfies γo(T ) ≥ (n − + s + 1)/3 and we characterize extremal trees attaining this lower bound. Then we give a constructive characterization of trees with equal domination and global offensive alliance numbers.

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Offensive alliances in graphs

Cited by 57 publications

References 3 publications

Partitioning a Graph into Global Powerful k-Alliances

Partitioning a Graph into Global Powerful k-Alliances

Expanding network communities from representative examples

On the global offensive alliance number of a tree

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