On the parameterized complexity of [1,j]-domination problems

For a graph G, a set D⊆V(G) is called a [1,j]-dominating set if every vertex in V(G)∖D has at least one and at most j neighbors in D. A set D⊆V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(TOTAL) DOMINATING SET problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-DOMINATING SET problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-TOTAL DOMINATING SET problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the DOMINATING SET problem, both are W[1]-hard when parameterized by solution size. In this work, we study the aforementioned problems on various graph classes from the perspective of parameterized complexity and prove the following results: • [1,j]-DOMINATING SET parameterized by solution size is W[1]-hard on d-degenerate graphs for d=j+1. • [1,j]-DOMINATING SET parameterized by solution size is FPT on nowhere dense graphs. • The known algorithm for [1,j]-DOMINATING SET on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH). • Assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-TOTAL DOMINATING SET problem parameterized by pathwidth. • Finally, we study another variant of DOMINATING SET, called RESTRAINED DOMINATING SET, that asks if there is a dominating set D of G of size at most k such that no vertex outside of D has all of its neighbors in D. We prove this variant is W[1]-hard even on 3-degenerate graphs