Domination and convexity problems in the target set selection model

In the target set selection model (TSS), introduced by Chen in 2009, it is given 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 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 time tτ(S0) of the target set S0. The problem TSS-Size of finding a minimum target set has been widely studied in the literature. In 2013, Cicalese et al. investigated the problem of finding a target set with activation time 1, which we call here TSS-Domination. In 2021, Keiler et al. investigated the problem TSS-Time of finding a target set with maximum activation time. In this paper, we introduce the problem of finding a maximum proper subset of vertices which is not a target set, which we call TSS-Max-Convex. We prove that, even in split graphs and bipartite graphs, TSS-max-convex is Poly-APX-Complete, TSS-Domination is Log-APX-Complete and their parameterized versions are W[1]-hard and W[2]-hard, respectively, when parameterized by the size of the solution and the maximum threshold τ∗=maxv∈V(G)τ(v). Moreover, we prove that both problems are FPT in bounded local-treewidth graphs when parameterized by the size of the solution and the maximum threshold τ∗. Finally, we prove that TSS-Domination is FPT in d-degenerate graphs when parameterized by the size of the solution and the degeneracy d, extending the main result of Alon and Gutner in 2009.