Irreversible conversion of graphs

Given a graph G, a function f:V(G)->Z, and an initial 0/1-vertex-labelling c"1:V(G)->{0,1}, we study an iterative 0/1-vertex-labelling process on G where in each round every vertex v never changes its label from 1 to 0, and changes its label from 0 to 1 if at least f(v) neighbours have label 1. Such processes model opinion/disease spreading or fault propagation and have been studied under names such as irreversible threshold/majority processes in a large variety of contexts. Our contributions concern computational aspects related to the minimum cardinality irr"f(G) of sets of vertices with initial label 1 such that during the process on G all vertices eventually change their label to 1. Such sets are known as irreversible conversion sets, dynamic irreversible monopolies, or catastrophic fault patterns. Answering a question posed by Dreyer and Roberts [P.A. Dreyer Jr., F.S. Roberts, Irreversible k-threshold processes: graph-theoretical threshold models of the spread of disease and of opinion, Discrete Appl. Math. 157 (2009) 1615-1627], we prove a hardness result for irr"f(G) where f(v)=2 for every [email protected]?V(G). Furthermore, we describe a general reduction principle for irr"f(G), which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs.