Target set selection with maximum activation time

A target set selection model is a graph G with a threshold function tau : V(G) -> N upper-bounded by the vertex degree. For a given model, a set S-0 subset of 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 all i is an element of {1, ... , t}, S-i contains exactly every vertex v having at least tau(v) neighbors in S-0 boolean OR ... boolean OR Si-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) >= k. Let tau* = max(nu is an element of v(G)) tau (nu). 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 tau*; otherwise, it is NP -complete even for fixed k = 4 and tau* = 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). (C) 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XI Latin and American Algorithms, Graphs and Optimization Symposium