# Target set selection with maximum activation time

A target set selection model is a graph G with a threshold function τ:V→ℕ upper-bounded by the vertex degree. For a given model, a set S_0⊆ V(G) is a target set if V(G) can be partitioned into non-empty subsets S_0,S_1,…,S_t such that, for i ∈{1, …, t}, S_i contains exactly every vertex v having at least τ(v) neighbors in S_0∪…∪ 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)≥ k. Let τ^* = max_v ∈ 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 C: if C has bounded local treewidth, the problem is FPT parameterized by k and τ^⋆; otherwise, it is NP-complete even for fixed k=4 and τ^⋆=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).

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