A note on power domination in grid graphs

Dorfling, Michael; Henning, Michael A.

doi:10.1016/j.dam.2005.08.006

Cited by 84 publications

(63 citation statements)

References 3 publications

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“…Dorfling and Henning computed the power domination number, i.e. the size of optimal power dominating set, for n × m grids [12]. Brueni and Heath [7] have more results on PDS, especially the NP-completeness of PDS on planar bipartite graphs.…”

Section: Previous Literaturementioning

confidence: 99%

Approximation Algorithms and Hardness for Domination with Propagation

Aazami¹,

Stilp²

2009

SIAM J. Discrete Math.

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The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or v has a neighbor in S, or (2) v has a neighbor w such that w and all of its neighbors except v are power dominated. We show a hardness of approximation threshold of 2 log 1−ǫ n in contrast to the logarithmic hardness for the dominating set problem. We give an O( √ n) approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of 2 log 1−ǫ n for directed acyclic graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.

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Section: Previous Literaturementioning

confidence: 99%

Approximation Algorithms and Hardness for Domination with Propagation

Aazami¹,

Stilp²

2009

SIAM J. Discrete Math.

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“…Some special classes of graphs have also been considered from an algorithmic point of view [3,5,10,11,19,26,27,36]. Dorfling and Henning [11] and Pai et al [26] determined the power domination number in grid graphs. Atkins et al [3], Hon et al [19], and Xu et al [36] proposed linear time algorithms for the power domination problem in block graphs.…”

Section: The Minimum Cardinality Of a Pds Of A Graph G Is Called The mentioning

confidence: 99%

Power Domination in Circular-Arc Graphs

Liao

Lee

2011

Algorithmica

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A set S ⊆ V is a power dominating set (PDS) of a graph G = (V, E) if every vertex and every edge in G can be observed based on the observation rules of power system monitoring. The power domination problem involves minimizing the cardinality of a PDS of a graph. We consider this combinatorial optimization problem and present a linear time algorithm for finding the minimum PDS of an interval graph if the interval ordering of the graph is provided. In addition, we show that the algorithm, which runs in Θ(n log n) time, where n is the number of intervals, is asymptotically optimal if the interval ordering is not given. We also show that the results hold for the class of circular-arc graphs.

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“…The set S is a power dominating set of G, PD-set for short, if M (S) = V (G). The power domination number γ P (G) of G is the minimum cardinality of a PD-set in G. Actually, the formulation of the power domination problem as just given is not the original definition but an equivalent simplification of it that was independently proposed in [8,9]. Recently, the power domination was extended to the so-called generalized power domination [5].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, linear-time algorithms for finding a minimum power dominating set were given for trees [13], for block graphs [27], and for interval graphs [22]. The exact value of the power domination number was determined for some products of graphs in [8,9], bounds for the power domination numbers of connected graphs and of claw-free cubic graphs are given in [31]. As already mentioned, the k-power domination was first studied in [5].…”

Section: Introductionmentioning

confidence: 99%

Generalized Power Domination: Propagation Radius and Sierpiński Graphs

Dorbec

Klavžar

2014

Acta Appl Math

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The recently introduced concept of k-power domination generalizes domination and power domination, the latter concept being used for monitoring an electric power system. The k-power domination problem is to determine a minimum size vertex subset S of a graph G such that after setting X = N [S], and iteratively adding to X vertices x that have a neighbour v in X such that at most k neighbours of v are not yet in X, we get X = V (G). In this paper the k-power domination number of Sierpiński graphs is determined. The propagation radius is introduced as a measure of the efficiency of power dominating sets. The propagation radius of Sierpiński graphs is obtained in most of the cases.

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A note on power domination in grid graphs

Cited by 84 publications

References 3 publications

Approximation Algorithms and Hardness for Domination with Propagation

Approximation Algorithms and Hardness for Domination with Propagation

Power Domination in Circular-Arc Graphs

Generalized Power Domination: Propagation Radius and Sierpiński Graphs

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