Burning Two Worlds

Kamali, Shahin; Miller, Avery; Zhang, Kenny

doi:10.1007/978-3-030-38919-2_10

Cited by 13 publications

(11 citation statements)

References 20 publications

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“…Since Theorem 10 and Theorem 11 follow as a result of Theorem 9, we are motivated to prove Theorem 9. The proof can be seen as a generalization of the proof of Theorem 1 in [10]. It should be noted that Theorem 9 illustrates an issue with the result in Theorem 1 of [10].…”

Section: Graded Families Of Graphsmentioning

confidence: 95%

“…For a vertex v, and a fixed positive integer r, the closed r-neighborhood of v, denoted N r (v), is the set of vertices of distance at most r from v. That is N r (v) = {u ∈ V (G) : d(u, v) ≤ r}. For a fixed positive integer r, closed r-neighborhoods of vertices inform an upper bound on the burning number through the following lemma, which encapsulates ideas from Section 2 of [5] but whose proof is in Section 2 of [10]. We include a proof for completeness.…”

Section: Preliminariesmentioning

confidence: 99%

“…Finally, we recall what is known about the burning number of graphs that have degree lower bounds. A breakthrough result in this direction was proven in [10].…”

Section: Corollarymentioning

confidence: 99%

“…The proof can be seen as a generalization of the proof of Theorem 1 in [10]. It should be noted that Theorem 9 illustrates an issue with the result in Theorem 1 of [10]. Therein, the authors use an instance of the framework here with m n,d = 3n 2(d+1) and evaluate their analogous function g n,d (x) at this value.…”

Section: Graded Families Of Graphsmentioning

confidence: 99%

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Burning Graph Classes

Omar¹,

Rohilla²

2021

Preprint

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The Burning Number Conjecture, that a graph on n vertices can be burned in at most ⌈ √ n ⌉ rounds, has been of central interest for the past several years. Much of the literature toward its resolution focuses on two directions: tightening a general upper bound for the burning number, and proving the conjecture for specific graph classes. In the latter, most of the developments work within a specific graph class and exploit the intricacies particular to it. In this article, we broaden this approach by developing systematic machinery that can be used as test beds for asserting that graph classes satisfy the conjecture. We show how to use these to resolve the conjecture for several classes of graphs including triangle-free graphs with degree lower bounds, graphs with certain linear lower bounds on r-neighborhood sizes, all trees whose nonleaf vertices have degree at least 4, trees whose non-leaf vertices have degree at least 3 (on at least 81 vertices), trees whose non-leaf vertices are less than 2 3 concentrated in degree 2, and trees with a low concentration of high degree non-leaf vertices (the last two results holding for sufficiently many non-leaf vertices).

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Section: Graded Families Of Graphsmentioning

confidence: 95%

Section: Preliminariesmentioning

confidence: 99%

“…Finally, we recall what is known about the burning number of graphs that have degree lower bounds. A breakthrough result in this direction was proven in [10].…”

Section: Corollarymentioning

confidence: 99%

Section: Graded Families Of Graphsmentioning

confidence: 99%

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Burning Graph Classes

Omar¹,

Rohilla²

2021

Preprint

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“…Graph burning was first studied in [6] with a notion that only one new fire source is initiated in each time step, and in each time-step the fire spreads as well. Graph burning has been shown to be NP-Complete in [4,22,23] and has been studied several times in [6,7,5,17,21,26,27,29,35]. Graph burning where more than one (but a constant number of) nodes has been studied in [33].…”

Section: Related Workmentioning

confidence: 99%

Spread and defend infection in graphs

Gupta¹

2021

Preprint

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The spread of an infection, a contagion, meme, emotion, message and various other spreadable objects have been discussed in several works. Burning and firefighting have been discussed in particular on static graphs. Burning simulates the notion of the spread of "fire" throughout a graph (plus, one unburned node burned at each time-step), graph firefighting simulates the defending of nodes by placing firefighters on the nodes which have not been already burned while the fire is being spread (started by a single fire source).This article studies a combination of firefighting and burning on a graph class which is a variation (generalization) of temporal graphs. Nodes can be infected from "outside" a network. We present a notion of both upgrading (of unburned nodes, similar to firefighting) and repairing (of infected nodes). The nodes which are burned, firefighted, or repaired are chosen probabilistically. So a variable amount of nodes are infected, upgraded and repaired in each time step.In the model presented in this article, both burning and firefighting proceed concurrently, we introduce such a system to enable the community to study the notion of spread of an infection and the notion of upgrade/repair against each other. The graph class that we study these processes is a variation of temporal graph class in which at each timestep, probabilistically, a communication takes place (iff an edge exists in that time step). In addition, a node can be "worn out" and thus can be removed from the network, a new healthy node can be added to the network. This class of graphs enables systems with high complexity to be able to be simulated and studied.

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