Abstract

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.

@InProceedings{alambardarmeybodi_et_al:LIPIcs:2018:9933,
  author =	{Mohsen Alambardar Meybodi and Fedor Fomin and Amer E. Mouawad and Fahad Panolan},
  title =	{{On the Parameterized Complexity of [1,j]-Domination Problems}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software  Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{34:1--34:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Sumit Ganguly and Paritosh Pandya},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9933},
  URN =		{urn:nbn:de:0030-drops-99330},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.34},
  annote =	{Keywords: [1, j]-dominating set, parameterized complexity, sparse graphs}
}