On the Parameterized Complexity of [1, j]-Domination Problems

Abstract

Abstract 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.