A linear algorithm for the domination number of a tree

Cockayne, E. J.; Goodman, Seymour E.; Hedetniemi, Stephen T.

doi:10.1016/0020-0190(75)90011-3

Cited by 163 publications

(62 citation statements)

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“…Slightly modifying the liner algorithm for the domination number of a tree designed by E Cockayne,S Goodman and S Hedetniemi Cock et al [1] ,which gives connected dominating tree. Let us consider the algorithm.…”

Section: Dominating Tree[2]mentioning

confidence: 99%

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Clustering Wireless Sensor and Wireless Ad Hoc Networks using Dominating Tree Concepts

Baidari¹,

Walikar²,

Shinde³

2012

IJCA

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The most important criterion for achieving the maximum performance in wirless sensor and ad-hoc networks is to clustering the nodes. Many clustering schemes have been proposed for different ad-hoc networks. In sensor networks the energy stored in the network nodes is limited and usually infeasible to recharge. The clustering schemes for these networks therefore aim at maximizing the energy efficiency. In mobile ad hoc networks the moment of the network nodes may quickly change the topology resulting in the increase of the overhead message in topology maintenance, the clustering schemes for mobile ad hoc networks therefore aim at handling topology maintenance, managing node movement or reducing overhead. In this paper we proposed an algorithm which gives connected dominating set of a tree.

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Section: Dominating Tree[2]mentioning

confidence: 99%

“…The following algorithm which computes value of for an arbitrary unweighted tree . It is taken from Mitchell, E. Cockayne and Hedetniemi [2] and is a linear implementation of the algorithm [1] …”

Section: Step11 [Process Last Vertex] If the Last Vertex Is Not Freementioning

confidence: 99%

Clustering Wireless Sensor and Wireless Ad Hoc Networks using Dominating Tree Concepts

Baidari¹,

Walikar²,

Shinde³

2012

IJCA

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“…In the special case r = s = 1, γ r, s; G is the well-studied domination number γ(G) [6]. A linear algorithm for γ(T ) was given in [3]. Other recently studied special cases are r = s = k (called {k}-domination [1,4]) and r = 1, s = k (termed k-tuple domination [1,5,7]).…”

Section: Introductionmentioning

confidence: 99%

An algorithm for the (r,s)-domination number of a tree

Cockayne

2007

ORiON

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Suppose that at most r units of some commodity may be positioned at any vertex of a graph G = (V, E) while at least s (≥ r) units must be present in the vicinity (i.e. closed neighbourhood) of each vertex. Suppose that the function f : V → {0, . . . , r}, whose values are the numbers of units stationed at vertices, satisfies the above requirement. Then f is called an s-dominating r-function. We present an algorithm which finds the minimum number of units required in such a function and a function which attains this minimum, for any tree.

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“…Since this problem is very hard and NP-complete even for special kinds of graphs such as planar graphs, much attention has focused on solving this problem on a more restricted class of graphs. It is well known that this problem can be solved on trees [CGH75] or even the generalization of trees, graphs of bounded treewidth [TP93]. The approximability of the dominating set problem has received considerable attention, but it is not known and it is not believed that this problem has constant factor approximation algorithms on general graphs [ACG + 99].…”

Section: Introductionmentioning

confidence: 99%

Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

2004

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Abstract. We present a fixed parameter algorithm that constructively solves the k-dominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1) ). In fact, we present our algorithm for any H-minor-free graph where H is a single-crossing graph (can be drawn on the plane with at most one crossing) and obtain the algorithm for K3,3(K5)-minor-free graphs as a special case. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set and a series of vertex removal problems. Our work generalizes and extends the recent result of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (non-planar) classes of graphs.

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A linear algorithm for the domination number of a tree

Cited by 163 publications

References 1 publication

Clustering Wireless Sensor and Wireless Ad Hoc Networks using Dominating Tree Concepts

Clustering Wireless Sensor and Wireless Ad Hoc Networks using Dominating Tree Concepts

An algorithm for the (r,s)-domination number of a tree

Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

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