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 [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. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms, Theory of computation → Design and analysis of algorithms