摘要：For a graph G, a set D subseteq V(G) is called a [1,j]-dominating set if every vertex in V(G) setminus D has at least one and at most j neighbors in D. A set D subseteq 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 [1,j]-Dominating Set on sparse graph classes from the perspective of parameterized complexity and prove the following results when the problem is parameterized by solution size:
- [1,j]-Dominating Set is W[1]-hard on d-degenerate graphs for d = j + 1; - [1,j]-Dominating Set is FPT on nowhere dense graphs. We also prove that the known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH). Finally, 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.
