Target set selection with maximum activation time

A target set selection model is a graph G with threshold function τ : V(G) → N upper-bounded by the vertex degree. For given model, S0 ⊆ if can be partitioned into non-empty subsets S0, S1,....,St such that, for all i∈{1,....,t}, Si contains exactly every v having at least τ(v) neighbors in S0∪⋯∪Si−1. We say that t activation time tτ(S0) of S0. The problem of, finding minimum size has been extensively studied literature. In this article, we investigate its variant, which call TSS-time, goal to find maximizes tτ(S0). That is, G, and an integer k, objective TSS-time decide whether tτ(S0)≥k. Let τ*=maxV∈V(G)τ(v). Our main result following dichotomy about complexity when belongs minor-closed class C: C bounded local treewidth, FPT parameterized k τ*; otherwise, it NP-complete even fixed = 4 τ* 2. also prove 2, NP-hard bipartite graphs 5, from previous results observe planar W[1]-hard treewidth. Finally, present linear-time algorithm tree maximizing