Irreversible 2-conversion set is NP-complete

Abstract

An irreversible k-threshold process is a process on a graph where vertices change color from white to black if they have at least k black neighbors. An irreversible k-conversion set is a subset S of vertices of a graph G such that the irreversible k-threshold process changes all vertices of G to black if S is the initial set of black vertices. We show that deciding the existence of an irreversible 2-conversion set of a given size is NP-complete which answers a question of Dreyer and Roberts. Moreover, we show an optimal irreversible 3-conversion set for a toroidal grid, which simplifies constructions of Pike and Zou.