Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree (2007.03959v3)

Abstract: We consider a non-monotone activation process $(X_t){t\in{ 0,1,2,\ldots}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t={ u\in V(G):|N_G(u)\cap X{t-1}|\geq \tau(u)}$ 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 $t\geq t_0$. 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.