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Topics in Graph Burning and Datalog

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posted on 25.05.2022, 17:10 by Daniel Moghbel

Graph burning studies how fast a contagion, modelled as a set of fires, spreads in a graph. The burning process takes place in discrete time-steps, or rounds. In each round, a fire breaks out at a given node (or vertex), thus burning it. Between rounds, fires from burning nodes spread to adjacent nodes. The burning number of a graph G, denoted b(G), is the minimum number of rounds necessary for every node of G to burn. We consider b(Gm,n), where Gm,n is the m × n Cartesian grid. For m = ω( √ n), the asymptotic value of b(Gm,n) was determined, but only the growth rate of b(Gm,n) was investigated in the case m = O( √ n). Accordingly, we provide new explicit bounds on b(Gc √ n,n) for valid  c > 0. Graph burning is analogous to a pebble game, which typically involves the placement of pebbles on nodes of a graph. Burning of a node is comparable to a pebbling step (or pebbling move): the removal of two pebbles from a node, where one of the removed pebbles is placed on an adjacent node while the other is discarded. In a certain pebble game  iii  variant (discussed in Section 1.7), the existence of a winning strategy has an interesting characterization: expressibility of the relevant constraint satisfaction problem (or CSP) in the logic programming language Datalog. If a structure with a non-empty domain is restricted to relation symbols only, then we call that structure a template. We show that the CSP for any finite template admitting terms of the weak J ́onsson type has a property known as bounded pathwidth duality. This implies the expressibility of the complement CSP in linear Datalog, and places the CSP in NL.


History

Language

English

Degree

Doctor of Philosophy

Program

Applied Mathematics

Granting Institution

Ryerson University

LAC Thesis Type

Dissertation

Thesis Advisor

Anthony Bonato Dejan Delic

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