Alambardar Meybodi, Mohsen and Fomin, Fedor V and Panolan, Fahad and et al, .
(2020)
On the parameterized complexity of [1,j]domination problems.
Theoretical Computer Science, 804.
pp. 207218.
ISSN 03043975
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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 NPcomplete, even for restricted classes of graphs such as chordal and planar graphs, but polynomialtime solvable on split graphs. The [1,2]TOTAL DOMINATING SET problem is known to be NPcomplete, 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 ddegenerate 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 3degenerate graphs.
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