A Study of Monopolies in Graphs

Khoshkhah, Kaveh; Nemati, Mehdi; Soltani, Hossein; Zaker, Manouchehr

doi:10.1007/s00373-012-1214-7

Cited by 17 publications

(10 citation statements)

References 13 publications

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“…We end this section with relating the majority strict dynamic monopolies to matching number of graphs. By the matching number of G we mean the maximum number of independent edges in G. In obtaining the next result we shall make use of a theorem from [13]. For this purpose we need some terminology.…”

Section: Remarkmentioning

confidence: 99%

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On dynamic monopolies of graphs: The average and strict majority thresholds

Khoshkhah

Soltani

Zaker

2012

Discrete Optimization

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Let G be a graph and τ : V (G) → N ∪ {0} be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be a dynamic monopoly corresponding to (G, τ ) if the vertices of G can be partitioned into subsets D 0 , D 1 , . . . , D k such that D 0 = D and for any i ∈ {0, . . . , k − 1}, each vertex v in D i+1 has at least τ (v) neighbors in D 0 ∪ . . . ∪ D i . Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of any dynamic monopoly in terms of the average threshold and the order of graph. Also we obtain an upper bound in terms of the minimum vertex cover of graphs. Then we derive the upper bound |G|/2 for the smallest size of any dynamic monopoly when the graph G contains at least one odd vertex, where the threshold of any vertex v is set as ⌈(deg(v) + 1)/2⌉ (i.e. strict majority threshold). This bound improves the best known bound for strict majority threshold. We show that the latter bound can be achieved by a polynomial time algorithm. We also show that α ′ (G) + 1 is an upper bound for the size of strict majority dynamic monopoly, where α ′ (G) stands for the matching number of G. Finally, we obtain a basic upper bound for the smallest size of any dynamic monopoly, in terms of the average threshold and vertex degrees. Using this bound we derive some other upper bounds.Mathematics Subject Classification: 91D30, 05C85, o5C69.

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Section: Remarkmentioning

confidence: 99%

“…By a graph parameter p we mean any function p from the set of all graphs to non-negative integers such that if G and H are two isomorphic graphs then p(G) = p(H). Also a graph parameter p is called subadditive if p(G ∪ H) ≤ p(G) + p(H), where G ∪ H is the vertex disjoint union of two graphs G and H. The following was proved in [13].…”

Section: Remarkmentioning

confidence: 99%

On dynamic monopolies of graphs: The average and strict majority thresholds

Khoshkhah

Soltani

Zaker

2012

Discrete Optimization

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“…Monopolies in graphs were defined first in [10] and they were generalized to k-monopolies recently in [9]. Other studies about monopolies in graphs and some of its applications can be found in [2,7,11,12,16].…”

Section: Introductionmentioning

confidence: 99%

Computing the (k-)monopoly number of direct product of graphs

Kuziak

Peterin

Yero

2015

Filomat

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Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ(G), and let k ∈ {1 − δ(G)/2 ,. .. , δ(G)/2 } be an integer. Given a set M ⊂ V, a vertex v of G is said to be k-controlled by M if δ M (v) ≥ δ(v) 2 + k where δ M (v) represents the quantity of neighbors v has in M and δ(v) the degree of v. The set M is called a k-monopoly if it k-controls every vertex v of G. The minimum cardinality of any k-monopoly is the k-monopoly number of G. In this article we study the k-monopoly number of direct product graphs. Specifically we obtain tight lower and upper bounds for the k-monopoly number of direct product graphs in terms of the k-monopoly numbers of its factors. Moreover, we compute the exact value for the k-monopoly number of several families of direct product graphs.

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“…For more details in dynamos in graphs (see [4,5,8,17]). In [14], the author defined a monopoly set of a graph G, proved that the mo(G) for general graph is at least n 2 , discussed the relationship between matchings and monopolies and he showed that any graph G admits a monopoly with at most α (G) vertices.…”

Section: Introductionmentioning

confidence: 99%

The Minimum Monopoly Distance Energy of a Graph

Naji¹,

Soner²

2015

IJCA

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In a graph G = (V, E), a set M ⊆ V is called a monopoly set of G if every vertex v ∈ V − M has at least d(v) 2 neighbors in M. The monopoly size mo(G) of G is the minimum cardinality of a monopoly set among all monopoly sets in G. In this paper, the minimum monopoly distance energy E M d (G) of a connected graph G is introduced and minimum monopoly distance energies of some standard graphs are computed. Some properties of the characteristic polynomial of the minimum monopoly distance matrix of G are obtained. Finally. Upper and lower bounds for E M d (G) are established.

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A Study of Monopolies in Graphs

Cited by 17 publications

References 13 publications

On dynamic monopolies of graphs: The average and strict majority thresholds

On dynamic monopolies of graphs: The average and strict majority thresholds

Computing the (k-)monopoly number of direct product of graphs

The Minimum Monopoly Distance Energy of a Graph

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