Target set selection with maximum activation time

A target set selection model is a graph $G$ with a threshold function $τ:V\to \mathbb{N}$ upper-bounded by the vertex degree. For a given model, a set $S_0\subseteq V(G)$ is a target set if $V(G)$ can be partitioned into non-empty subsets $S_0,S_1,\dotsc,S_t$ such that, for $i \in \{1, \ldots, t\}$, $S_i$ contains exactly every vertex $v$ having at least $τ(v)$ neighbors in $S_0\cup\dots\cup S_{i-1}$. We say that $t$ is the activation time $t_τ(S_0)$ of the target set $S_0$. The problem of, given such a model, finding a target set of minimum size has been extensively studied in the literature. In this article, we investigate its variant, which we call TSS-time, in which the goal is to find a target set $S_0$ that maximizes $t_τ(S_0)$. That is, given a graph $G$, a threshold function $τ$ in $G$, and an integer $k$, the objective of the TSS-time problem is to decide whether $G$ contains a target set $S_0$ such that $t_τ(S_0)\geq k$. Let $τ^* = \max_{v \in V(G)} τ(v)$. Our main result is the following dichotomy about the complexity of TSS-time when $G$ belongs to a minor-closed graph class ${\cal C}$: if ${\cal C}$ has bounded local treewidth, the problem is FPT parameterized by $k$ and $τ^{\star}$; otherwise, it is NP-complete even for fixed $k=4$ and $τ^{\star}=2$. We also prove that, with $τ^*=2$, the problem is NP-hard in bipartite graphs for fixed $k=5$, and from previous results we observe that TSS-time is NP-hard in planar graphs and W[1]-hard parameterized by treewidth. Finally, we present a linear-time algorithm to find a target set $S_0$ in a given tree maximizing $t_τ(S_0)$.