Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

Gentner, Michael; Rautenbach, Dieter

doi:10.1016/j.tcs.2016.12.028

Cited by 7 publications

(12 citation statements)

References 22 publications

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“…In particular, for connected graphs of order n > 1∕(2 ) we always have h (G) < 2 n. The second main result of this paper is a stronger bound for graphs of girth at least five, obtained by combining our approach for Theorem 1.2 with the method used by Gentner and Rautenbach [29] to prove (4). Theorem 1.3 For every > 0 there exists 0 > 0 such that for every ∈ (0, 0 ) and every connected graph G of order n and girth at least 5 we have h (G) = 1 or h (G) < (1 + ) n. Theorem 1.3 is also asymptotically best possible, as shown by the following example of the balanced (⌊1∕ ⌋ + 1)-regular tree.…”

Section: Introductionmentioning

confidence: 88%

“…To prove our second main theorem we use the following modified version of a theorem of Gentner and Rautenbach . For a constant ρ > 0 and a graph G we again use the notation V 1 = { v ∈ V ( G )| d ( v ) ≤ 1/ ρ } and V ≥2 = V ∖ V 1 .…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…The form of the lemma given above follows immediately from the proof of the original version [29,Theorem 1], which omitted the condition on the neighborhood in V 1 of vertices in V ≥2 , and had the weaker conclusion that h (G) ≤ (2 + ) n. Loosely speaking, in this proof Gentner and Rautenbach used a random selection process to select an initial set X and showed that vertices satisfying certain properties are infected with high probability by X. They then bounded the number of vertices which do not have these properties, and simply added all such vertices to X to obtain a contagious set of the claimed size.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…The first bound of the form described above was due to Chang , who showed that every connected graph G on n vertices satisfies h ρ ( G ) = 1 or

h_{ρ} false(G false) \leq false(2 \sqrt{2} + 3 false) ρ n < 5.83 ρ n .

Chang and Lyuu improved this bound by showing that every such G satisfies h ρ ( G ) = 1 or

h_{ρ} (G) \leq 4.92 ρ n .

This was the best general bound prior to this work. Very recently Gentner and Rautenbach provided an upper bound which asymptotically matches the lower bound of Observation under the additional assumptions that G has girth at least five and ρ is sufficiently small. More precisely, they showed that for any ε > 0, if ρ is sufficiently small and G has girth at least five, then h ρ ( G ) = 1 or

h_{ρ} false(G false) \leq false(2 + ε false) ρ n .

The first main result of this paper is the following theorem, which improves on each of the aforementioned results by giving the optimal bound of the described form.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, for connected graphs of order n > 1/(2 ρ ) we always have h ρ ( G ) < 2 ρn . The second main result of this paper is a stronger bound for graphs of girth at least five, obtained by combining our approach for Theorem with the method used by Gentner and Rautenbach to prove .…”

Section: Introductionmentioning

confidence: 99%

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Contagious sets in a degree‐proportional bootstrap percolation process

Garbe

Mycroft

McDowell

2018

Random Struct Algorithms

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We study the following bootstrap percolation process: given a connected graph G, a constant ρ ∈ [0,1] and an initial set A⊆V(G) of infected vertices, at each step a vertex v becomes infected if at least a ρ‐proportion of its neighbors are already infected (once infected, a vertex remains infected forever). Our focus is on the size hρ(G) of a smallest initial set which is contagious, meaning that this process results in the infection of every vertex of G. Our main result states that every connected graph G on n vertices has hρ(G) < 2ρn or hρ(G) = 1 (note that allowing the latter possibility is necessary because of the case ρ≤12n, as every contagious set has size at least one). This is the best‐possible bound of this form, and improves on previous results of Chang and Lyuu and of Gentner and Rautenbach. We also provide a stronger bound for graphs of girth at least five and sufficiently small ρ, which is asymptotically best‐possible.

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Section: Introductionmentioning

confidence: 88%

Section: Proofs Of the Theoremsmentioning

confidence: 99%

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…The first bound of the form described above was due to Chang , who showed that every connected graph G on n vertices satisfies h ρ ( G ) = 1 or

h_{ρ} false(G false) \leq false(2 \sqrt{2} + 3 false) ρ n < 5.83 ρ n .

Chang and Lyuu improved this bound by showing that every such G satisfies h ρ ( G ) = 1 or

h_{ρ} (G) \leq 4.92 ρ n .

h_{ρ} false(G false) \leq false(2 + ε false) ρ n .

The first main result of this paper is the following theorem, which improves on each of the aforementioned results by giving the optimal bound of the described form.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Contagious sets in a degree‐proportional bootstrap percolation process

Garbe

Mycroft

McDowell

2018

Random Struct Algorithms

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Partial immunization of trees

Dourado

Ehard

Penso

et al. 2020

Discrete Optimization

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For a graph G and an integer-valued function τ on its vertex set, a dynamic monopoly is a set of vertices of G such that iteratively adding to it vertices u of G that have at least τ (u) neighbors in it eventually yields the vertex set of G. We study the problem of maximizing the minimum order of a dynamic monopoly by increasing the threshold values of individual vertices subject to vertex-dependent lower and upper bounds, and fixing the total increase. We solve this problem efficiently for trees, which extends a result of Khoshkhah and Zaker (On the largest dynamic monopolies of graphs with a given average threshold, Canadian Mathematical Bulletin 58 (2015) 306-316). IntroductionAs a simple model for an infection process within a network [12,13,16] one can consider a graph G in which each vertex u is assigned a non-negative integral threshold value τ (u) quantifying how many infected neighbors of u are required to spread the infection to u. In this setting, a dynamic monopoly of (G, τ ) is a set D of vertices such that an infection starting in D spreads to all of G, and the smallest order dyn(G, τ ) of such a dynamic monopoly measures the vulnerability of G for the given threshold values.Khoshkhah and Zaker [17] consider the maximum of dyn(G, τ ) over all choices for the function τ such that the average threshold is at most some positive realτ . They show that this maximum equals max k :

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