α-Domination

In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of α-domination and k-tuple domination.

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“…However, the probabilistic construction used to obtain the bounds (9) and (11) is different from that to obtain the bounds (4) and (7).…”

Section: Corollary 1 Generalises the Following Well-known Upper Boundmentioning

confidence: 99%

Upper Bounds for α-Domination Parameters

Gagarin

Poghosyan

Zverovich

2009

Graphs and Combinatorics

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“…Our first lemma ensures that i (G) is well defined for every graph G. This statement actually follows easily from a colouring result of Cowen and Emerson (stated as Theorem 8 in [3]). Nevertheless, in order to make our presentation self-contained we include a short proof using a classical Erdős-type exchange argument.…”

Section: Introductionmentioning

confidence: 70%

“…In the present paper we initiate the study of a similarly defined class of 'perfect' graphs using the concept of -domination that has recently been introduced by Dunbar et al [3]. Our main result is the characterization of -domination perfect trees.…”

Section: Introductionmentioning

confidence: 99%

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α-Domination perfect trees

Dahme

Rautenbach

Volkmann

2008

Discrete Mathematics

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Let ∈ (0, 1) and let G = (V G , E G ) be a graph. According to Dunbar et al. [ -Domination, Discrete Math. 211 (2000) the minimum cardinality of an -dominating set of G and the -independent -domination number i (G) of G is the minimum cardinality of an -dominating set of G that is also -independent. A graph G is -domination perfect if (H ) = i (H ) for all induced subgraphs H of G.We characterize the -domination perfect trees in terms of their minimally forbidden induced subtrees. For ∈ (0, 1 2 ] there is exactly one such tree whereas for ∈ ( 1 2 , 1) there are infinitely many.

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“…In graph theory such a set is called a dominating set and the problem of finding a dominating set of minimal cardinality is NP-complete [3]. The notion was generalised introducing k-domination where each node needs to have at least k neighbours in the dominating set, and α domination where 0 < α ≤ 1, where each node not in the dominating set needs to have at least α * 100 percent of its neighbours in the dominating set [4], and α-rate domination [5] where each node (including ones in the dominating set) needs to have at least α * 100 percent of its neighbours in the dominating set. Again, finding minimum cardinalities of α and α-rate dominating sets is NP-complete.…”

Section: Introductionmentioning

confidence: 99%

Weighted Alpha-Rate Dominating Sets in Social Networks

Greetham

Poghosyan

Charlton

2014

2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems

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We are looking into variants of a domination set problem in social networks. While randomised algorithms for solving the minimum weighted domination set problem and the minimum alpha and alpha-rate domination problem on simple graphs are already present in the literature, we propose here a randomised algorithm for the minimum weighted alpha-rate domination set problem which is, to the best of our knowledge, the first such algorithm. A theoretical approximation bound based on a simple randomised rounding technique is given. The algorithm is implemented in Python and applied to a UK Twitter mentions networks using a measure of individuals' influence (klout) as weights. We argue that the weights of vertices could be interpreted as the costs of getting those individuals on board for a campaign or a behaviour change intervention. The minimum weighted alpha-rate dominating set problem can therefore be seen as finding a set that minimises the total cost and each individual in a network has at least alpha percentage of its neighbours in the chosen set. We also test our algorithm on generated graphs with several thousand vertices and edges. Our results on this real-life Twitter networks and generated graphs show that the implementation is reasonably efficient and thus can be used for real-life applications when creating social network based interventions, designing social media campaigns and potentially improving users' social media experience.

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α-Domination

Cited by 39 publications

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Upper Bounds for α-Domination Parameters

Upper Bounds for α-Domination Parameters

α-Domination perfect trees

Weighted Alpha-Rate Dominating Sets in Social Networks

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