  A target set selection model is a graph $G$ with a threshold function
$\tau:V\to \mathbb{N}$ 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 $\tau(v)$ neighbors in
$S_0\cup\dots\cup S_{i-1}$. We say that $t$ is the activation time
$t_{\tau}(S_0)$ 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
$t_{\tau}(S_0)$. That is, given a graph $G$, a threshold function $\tau$ 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 $t_{\tau}(S_0)\geq k$. Let
$\tau^* = \max_{v \in V(G)} \tau(v)$. 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 $\tau^{\star}$; otherwise, it is NP-complete
even for fixed $k=4$ and $\tau^{\star}=2$. We also prove that, with $\tau^*=2$,
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 $t_{\tau}(S_0)$.
