Fabrício Benevides

Maximum Percolation Time in Two-Dimensional Bootstrap Percolation

We consider a classic model known as bootstrap percolation on the

The maximum time of 2-neighbour bootstrap percolation

n \times n

Edge-colorings of graphs avoiding complete graphs with a prescribed coloring

square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least two already infected neighbors. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is

The maximum infection time in the geodesic and monophonic convexities

13n^2/18+O(n)

Connected Greedy Colourings

.

Identifying defective sets using queries of small size

In 2-neighbourhood bootstrap percolation on a graph G , an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper, we are interested to calculate the maximal time t ( G ) the process can take, in terms of the number of times the interval function is applied, to eventually infect the entire vertex set. We prove that the problem of deciding if t ( G ) ? k is NP-complete for: (a) fixed k ? 4 ; (b) bipartite graphs and fixed k ? 7 ; and (c) planar graphs. Moreover, we obtain linear and polynomial time algorithms for trees and chordal graphs, respectively.

Circular backbone colorings: On matching and tree backbones of planar graphs

Given a graph

Complexity of determining the maximum infection time in the geodetic convexity

F

On Slowly Percolating Sets of Minimal Size in Bootstrap Percolation

and an integer

Counting Gallai 3-colorings of complete graphs

r \ge 2