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

TL;DR

This paper studies a non-monotone activation process on graphs with thresholds restricted to 0, 1, or the vertex degree, providing characterizations and complexity results for finding minimal target sets.

Contribution

It answers an open question by showing the minimum target set problem is efficiently solvable on trees for specific thresholds and characterizes the problem's complexity on various graph classes.

Findings

Efficient algorithms for target set selection on trees with restricted thresholds.

NP-hardness of the problem on planar graphs with maximum degree 3.

Polynomial-time solvability on graphs of bounded treewidth.

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.