Burning and w-burning of geometric graphs

Abstract

Abstract: Graph burning runs on discrete time-steps. The aim is to burn all the vertices in a given graph using a minimum number of time-steps. This number is known to be the burning number of the graph. The spread of social influence, an alarm, or a social contagion can be modeled using graph burning. The less the burning number, the faster the spread.

It is well-known that the optimal burning of general graphs is NP-complete. Further, graph burning has been shown to be NP-complete on a vast majority classes of graphs. Approximation results also exist for several graph classes. In this article, we show that the burning problem is NP-complete on connected interval graphs and permutation graphs. We also study the burning properties of grids. More precisely, we show that the lower bound of the burning number of a grid is at least . We provide a 2-approximation for burning a square grid.

We extend the study of the -burning problem, a variation of the graph burning problem where we allow a constant number of vertices to be burnt in any time-step. We prove that -burning of interval, spider, and permutation graphs are NP-complete for any constant . We also provide a 2-approximation for the -burning problem on trees.