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.
Highlights
A dominating set of a graph G is a subset D of vertices such that each vertex in V (G) \ D is adjacent to at least one vertex in D
We prove that the known algorithm for [1, j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH)
The authors in [3] raised several open problems, including whether restricting to specific classes of graphs leads to strictly better upper bounds for the size of [1, j]-dominating sets and whether [1, j]-Dominating Set is efficiently solvable on trees
Summary
A dominating set of a graph G is a subset D of vertices such that each vertex in V (G) \ D is adjacent to at least one vertex in D. For a constant j, a polynomial-time algorithm running in time O(njp(lg n)) where p is a polynomial function, was obtained for n-vertex split graphs [2] This is in contrast to the classic Dominating Set problem which is NP-hard for this class of graphs. To the best of our knowledge, this is the largest class of graphs for which the Dominating Set problem is known to be fixed-parameter tractable; d-degenerate and nowhere dense graphs are subclasses of t-biclique-free graphs. Another variant of dominating sets and [1, j]-dominating sets is [1, j]-total dominating sets.
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