Target set selection with maximum activation time

• Published in: Discrete Applied Mathematics Vol. 338; pp. 199 - 217
• Main Authors: Keiler, Lucas, Lima, Carlos Vinicius Gomes Costa, Maia, Ana Karolinna, Sampaio, Rudini, Sau, Ignasi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 30.10.2023; Elsevier
• Subjects: Activation time; Bipartite graph; Bounded local treewidth; Complexity dichotomy; Computer Science; Fixed-parameter tractability; Mathematics; Planar graph; Target set selection; Tree; Complexity dichotomy; Activation time; Planar graph; Tree; Target set selection; Bipartite graph; Fixed-parameter tractability; Bounded local treewidth; bounded local treewidth; activation time; tree; complexity dichotomy; fixed-parameter tractability; planar graph; bipartite graph
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2023.06.004

A target set selection model is a graph G with a threshold function τ:V(G)→N upper-bounded by the vertex degree. For a given model, a set S0⊆V(G) is a target set if V(G) can be partitioned into non-empty subsets S0,S1,…,St such that, for all i∈{1,…,t}, Si contains exactly every vertex v outside S0∪⋯∪Si−1 having at least τ(v) neighbors in S0∪⋯∪Si−1. We say that t is the activation timetτ(S0) of the target set S0. 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 S0 that maximizes tτ(S0). 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 S0 such that tτ(S0)≥k. Let τ⋆=maxv∈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 S0 in a given tree maximizing tτ(S0).