Abstract
In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarsky et al. [ESA 2013] proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a ctreedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c for a fixed integer c>0. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs.
In this article we answer this question by finding two very natural such problems: we prove that VERTEX COVER admits a polynomial kernel on general graphs for any integer c>0, and that DOMINATING SET does not for any integer c>1 even on degenerate graphs, unless NP is a subset of coNP/poly. For the positive result, we build on the techniques of Jansen and Bodlaender [STACS 2011], and for the negative result we use a polynomial parameter transformation for c>2 and an ORcrosscomposition for c=2. As existing results imply that DOMINATING SET admits a polynomial kernel on degenerate graphs for c=1, our result provides a dichotomy about the existence of polynomial problems for DOMINATING SET on degenerate graphs with this parameter.
BibTeX  Entry
@InProceedings{bougeret_et_al:LIPIcs:2018:8556,
author = {Marin Bougeret and Ignasi Sau},
title = {{How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphsl}},
booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
pages = {10:110:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770514},
ISSN = {18688969},
year = {2018},
volume = {89},
editor = {Daniel Lokshtanov and Naomi Nishimura},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8556},
URN = {urn:nbn:de:0030drops85564},
doi = {10.4230/LIPIcs.IPEC.2017.10},
annote = {Keywords: parameterized complexity, polynomial kernels, structural parameters, treedepth, treewidth, sparse graphs}
}
Keywords: 

parameterized complexity, polynomial kernels, structural parameters, treedepth, treewidth, sparse graphs 
Collection: 

12th International Symposium on Parameterized and Exact Computation (IPEC 2017) 
Issue Date: 

2018 
Date of publication: 

02.03.2018 