Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree

Abstract

We consider a non-monotone activation process $(X_t{)}_{t\in \{0,1,2,\dots \}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{u\in V(G):|N_G(u)\cap X_{t-1}|\ge \tau (u)\}$ for every positive integer $t$, and $\tau :V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau )$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\ge t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $\tau$ satisfying $\tau (u)\in \{0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.