Topics in Graph Burning and Datalog

Abstract

<p>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.</p> <p><br></p>