Abstract
A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. It is wellknown that the problem of finding a minimum sized (or k sized in case of decision version of) feedback vertex set (FVS) is polynomial time solvable in (sub)cubic graphs, in pseudoforests (graphs where each component has at most one cycle) and mockforests (graph where each vertex is part of at most one cycle). In general graphs, it is known that the problem is NPComplete, and has an O*((3.619)^k) fixedparameter algorithm and an O(k^2) kernel where k, the solution size is the parameter. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure of the input. In particular, we show that
* FVS is fixedparameter tractable, but is unlikely to have polynomial sized kernel when parameterized by the number of vertices of the graph whose degree is at least 4. This answers a question asked in an earlier paper.
* When parameterized by k, the number of vertices, whose deletion results in a pseudoforest, we give an O(k^6) vertices kernel improving from the previously known O(k^{10}) bound.
* When parameterized by the number k of vertices, whose deletion results in a mockdforest, we give a kernel consisting of O(k^{3d+3}) vertices and prove a lower bound of Omega(k^{d+2}) vertices (under complexity theoretic assumptions). Mockdforest for a constant d is a mockforest where each component has at most d cycles.
BibTeX  Entry
@InProceedings{majumdar:LIPIcs:2017:6933,
author = {Diptapriyo Majumdar},
title = {{Structural Parameterizations of Feedback Vertex Set}},
booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)},
pages = {21:121:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770231},
ISSN = {18688969},
year = {2017},
volume = {63},
editor = {Jiong Guo and Danny Hermelin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/6933},
URN = {urn:nbn:de:0030drops69337},
doi = {10.4230/LIPIcs.IPEC.2016.21},
annote = {Keywords: Parameterized Complexity, Kernelization, Feedback Vertex Set, Structural Parameterization}
}
Keywords: 

Parameterized Complexity, Kernelization, Feedback Vertex Set, Structural Parameterization 
Collection: 

11th International Symposium on Parameterized and Exact Computation (IPEC 2016) 
Issue Date: 

2017 
Date of publication: 

09.02.2017 