Parameterized complexity of weighted target set selection

Consider a graph G where each vertex has a threshold. A vertex v in G is activated if the number of active vertices adjacent to v is at least as many as its threshold. A vertex subset A 0 of G is a target set if eventually all vertices in G are activated by initially activating vertices of A 0 . The Target Set Selection problem ( TSS ) involves finding a smallest target set of G . This problem has already been extensively studied and is known to be NP-hard even for very restricted conditions. In this paper, we analyze TSS and its weighted variant, called the Weighted Target Set Selection problem ( WTSS ), from the perspective of parameterized complexity. Let k be the solution size and let ℓ be the maximum threshold. We first show that TSS is W[1]-hard for split graphs when parameterized by k + ℓ , and W[2]-hard for cographs when parameterized by k . We next prove that WTSS is W[2]-hard for trivially perfect graphs when parameterized by k . On the other hand, we show that WTSS can be solved in O ( n log ⁡ n ) time for complete graphs with n vertices. Additionally, we design FPT algorithms for WTSS when parameterized by nd + ℓ , tw + ℓ , ce , and vc , where nd , tw , ce , and vc are the neighborhood diversity, the treewidth, the cluster editing number, and the vertex cover number of the input graph, respectively.