Abstract
For nonempty σ, ρ ⊆ ℕ, a vertex set S in a graph G is a (σ, ρ)dominating set if for all v ∈ S, N(v) ∩ S ∈ σ, and for all v ∈ V(G) ⧵ S, N(v) ∩ S ∈ ρ. The Min/Max (σ,ρ)Dominating Set problems ask, given a graph G and an integer k, whether G contains a (σ, ρ)dominating set of size at most k and at least k, respectively. This framework captures many wellstudied graph problems related to independence and domination. BuiXuan, Telle, and Vatshelle [TCS 2013] showed that for finite or cofinite σ and ρ, the Min/Max (σ,ρ)Dominating Set problems are solvable in XP time parameterized by the mimwidth of a given branch decomposition of the input graph. In this work we consider the parameterized complexity of these problems and obtain the following: For minimization problems, we complete several scattered W[1]hardness results in the literature to a full dichotomoy into polynomialtime solvable and W[1]hard cases, and for maximization problems we obtain the same result under the additional restriction that σ and ρ are finite sets. All W[1]hard cases hold assuming that a linear branch decomposition of bounded mimwidth is given, and with the solution size being an additional part of the parameter. Furthermore, for all W[1]hard cases we also rule out f(w)n^o(w/log w)time algorithms assuming the Exponential Time Hypothesis, where f is any computable function, n is the number of vertices and w the mimwidth of the given linear branch decomposition of the input graph.
BibTeX  Entry
@InProceedings{bakkane_et_al:LIPIcs.IPEC.2022.3,
author = {Bakkane, Brage I. K. and Jaffke, Lars},
title = {{On the Hardness of Generalized Domination Problems Parameterized by MimWidth}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {3:13:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772600},
ISSN = {18688969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17359},
URN = {urn:nbn:de:0030drops173597},
doi = {10.4230/LIPIcs.IPEC.2022.3},
annote = {Keywords: generalized domination, linear mimwidth, W\lbrack1\rbrackhardness, Exponential Time Hypothesis}
}
Keywords: 

generalized domination, linear mimwidth, W[1]hardness, Exponential Time Hypothesis 
Collection: 

17th International Symposium on Parameterized and Exact Computation (IPEC 2022) 
Issue Date: 

2022 
Date of publication: 

14.12.2022 