How to Burn a Graph

Bonato, Anthony; Janssen, Jeannette; Roshanbin, Elham

doi:10.1080/15427951.2015.1103339

Cited by 70 publications

(123 citation statements)

References 23 publications

(32 reference statements)

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“…The burning number [5,6] measures how prone a network is to fast social contagion. In the burning protocol, like many other network protocols, data is communicated between nodes in discrete rounds.…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Graph Burning

Bonato

Kamali

2019

Lecture Notes in Computer Science

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Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5.We present approximation algorithms for graph burning. For general graphs, we introduce an algorithm with an approximation ratio of 3. When the graph is a tree, we present another algorithm with approximation ratio 2. Moreover, we consider a setting where the graph is a forest of disjoint paths. In this setting, when the number of paths is constant, we provide an optimal algorithm which runs in polynomial time. When the number of paths is more than a constant, we provide two approximation schemes: first, under a regularity condition where paths have asymptotically equal lengths, we show the problem admits an approximation scheme which is fully polynomial. Second, for a general setting where the regularity condition does not necessarily hold, we provide another approximation scheme which runs in time polynomial in the size of the graph.

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Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Graph Burning

Bonato

Kamali

2019

Lecture Notes in Computer Science

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“…In this paper, we study a graph parameter, the burning number, that was recently introduced as a simple model of spreading social influence [6]. Our focus is on a few probabilistic aspects of this problem.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Let us start with the original process introduced recently in [6]. Given a finite, simple, undirected graph G, the burning process on G is a discrete-time process defined as follows.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…. , v n }, with v i being adjacent to v i−1 for 2 ≤ i ≤ n. It is easy to see that b(P n ) = ⌈ √ n⌉ (see [6] for this and more results). The following result shows that the first two costs of drunkenness (c 1 (P n ) and c 2 (P n )) are asymptotically equal to each other, and that both b 1 (P n ) and b 2 (P n ) are much bigger than b(P n ).…”

Section: 3mentioning

confidence: 99%

“…We also deal with the Cartesian product of two paths, which is the only deterministic result shown in this paper. In [6], the burning number of the Cartesian product of two paths was studied, including non-symmetric cases, that is, the product of two paths of different lengths. However, only the order of this graph parameter was found.…”

Section: 3mentioning

confidence: 99%

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Burning Graphs: A Probabilistic Perspective

Mitsche¹,

Prałat

Roshanbin

2017

Graphs and Combinatorics

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Abstract. In this paper, we study a graph parameter that was recently introduced, the burning number, focusing on a few probabilistic aspects of the problem. The original burning number is revisited and analyzed for binomial random graphs G(n, p), random geometric graphs, and the Cartesian product of paths. Moreover, new variants of the burning number are introduced in which a burning sequence of vertices is selected according to some probabilistic rules. We analyze these new graph parameters for paths.

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