Rudini Menezes Sampaio

The maximum time of 2-neighbour bootstrap percolation

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.

Graphs with few P 4 's under the convexity of paths of order three

A graph is ( q , q - 4 ) if every subset of at most q vertices induces at most q - 4 P 4 s. It therefore generalizes some different classes, as cographs and P 4 -sparse graphs. In this work, we propose algorithms for determining various NP-Hard graph convexity parameters within the convexity of paths of order three, for ( q , q - 4 ) graphs. All algorithms have linear-time complexity, for fixed q , and then are fixed parameter tractable. Moreover, we prove that the Caratheodory number is at most three for every cograph, P 4 -sparse graph and every connected ( q , q - 4 ) -graph with at least q vertices.

On the Complexity of Solving or Approximating Convex Recoloring Problems

Given a graph with an arbitrary vertex coloring, the Convex Recoloring Problem (CR) consists of recoloring the minimum number of vertices so that each color induces a connected subgraph. We focus on the complexity and inapproximabiliy of this problem on k-colored graphs, for fixed k ≥ 2. We prove a very strong complexity result showing that CR is already NP-hard on k-colored grids, and therefore also on planar graphs with maximum degree 4. For each k ≥ 2, we also prove that, for a positive constant c, there is no cln n-approximation algorithm even for k-colored n-vertex bipartite graphs, unless P = NP. For 2-colored (q,q − 4)-graphs, a class that includes cographs and P 4-sparse graphs, we present polynomial-time algorithms for fixed q. The same complexity results are obtained for a relaxation of CR, where only one fixed color is required to induce a connected subgraph.

The Maximum Time of 2-Neighbour Bootstrap Percolation: Complexity Results

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. The maximum time \(t(G)\) is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved [7] that deciding if \(t(G)\ge k\) is polynomial time solvable for \(k=2\), but is NP-Complete for \(k=4\) and is NP-Complete if the graph is bipartite and \(k=7\). In this paper, we solve the open questions. Let \(n = |V(G)|\) and \(m = |E(G)|\). We obtain an \(\varTheta (m n^5)\)-time algorithm to decide if \(t(G)\ge 3\) in general graphs. In bipartite graphs, we obtain an \(\varTheta (m n^3)\)-time algorithm to decide if \(t(G)\ge 3\) and an \(O(m n^{13})\)-time algorithm to decide if \(t(G)\ge 4\). We also prove that deciding if \(t(G)\ge 5\) is NP-Complete in bipartite graphs.

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

Given a graph

Inapproximability results for graph convexity parameters

F

Inapproximability results related to monophonic convexity

and an integer

The convexity of induced paths of order three

r \ge 2

A note on permutation regularity

, a partition

Restricted coloring problems on graphs with few P4′s

\widehat{F}