Spectral study of alliances in graphs

Rodríguez-Velázquez, Juan A.; Almira, José María Sigarreta

doi:10.7151/dmgt.1351

Cited by 27 publications

(11 citation statements)

References 3 publications

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“…A set S ⊂ V is a k-dominating set if for every v ∈S, δ S (v) ≥ k. The k-domination number of G, γ k (G), is the minimum cardinality of a k-dominating set in G. The following result generalizes, to r alliances, some previous results obtained for r = 1 and r = 2 [14,16]. Theorem 6.…”

Section: Bounding the Offensive R-alliance Numbermentioning

confidence: 84%

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Offensive r-alliances in graphs

Fernau

Rodríguez

Sigarreta

2009

Discrete Applied Mathematics

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a b s t r a c tLet G = (V , E) be a simple graph. For a nonempty set X ⊂ V , and a vertex vthe boundary of S. An offensive ralliance S is called global if it forms a dominating set. The global offensive r-alliance number of G, denoted by γ o r (G), is the minimum cardinality of a global offensive r-alliance in G. Weshow that the problem of finding optimal (global) offensive r-alliances is NP-complete and we obtain several tight bounds on γ o r (G).

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Section: Bounding the Offensive R-alliance Numbermentioning

confidence: 84%

“…The reader is referred to our previous works [14,15,20,16] for a more detailed study on offensive 1-alliances and offensive 2-alliances.…”

Section: Remark 10mentioning

confidence: 99%

Offensive r-alliances in graphs

Fernau

Rodríguez

Sigarreta

2009

Discrete Applied Mathematics

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“…[7,13] Let Γ be a simple graph of order n and minimum degree δ. Let µ be the Laplacian spectral radius of Γ. Then…”

Section: We Only Need To Show Thatmentioning

confidence: 99%

“…The particular case of global (strong) defensive alliance was investigated in [5] where several bounds on the global (strong) defensive alliance number were obtained. In [7] there were obtained several tight bounds on different types of alliance numbers of a graph, namely the (global) defensive alliance number, (global) offensive alliance number and (global) dual alliance number. In particular, there was investigated the relationship between the alliance numbers of a graph and its algebraic connectivity, its spectral radius, and its Laplacian spectral radius.…”

Section: Introductionmentioning

confidence: 99%

Offensive alliances in cubic graphs

Rodríguez¹,

Sigarreta²

2006

imf

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An offensive alliance in a graph Γ = (V, E) is a set of vertices S ⊂ V where for every vertex v in its boundary it holds that the majority of vertices in v's closed neighborhood are in S. In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V \S, that is, S is a dominating set of Γ. The global offensive alliance number γ o (Γ) (respectively, global strong offensive alliance number γô(Γ)) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in Γ. If Γ has global independent offensive alliances, then the global independent offensive alliance number γ i (Γ) is the minimum cardinality among all independent global offensive alliances of Γ. In this paper we study mathematical properties of the global (strong) alliance number of cubic graphs. For instance, we show that for all connected cubic graph of order n,

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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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Spectral study of alliances in graphs

Cited by 27 publications

References 3 publications

Offensive r-alliances in graphs

Offensive r-alliances in graphs

Offensive alliances in cubic graphs

Partitioning a Graph into Global Powerful k-Alliances

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