Abstract
For a family of graphs F, a graph G, and a positive integer k, the FDELETION problem asks whether we can delete at most k vertices from G to obtain a graph in F. FDELETION generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph G = (V, cup_{i=1}^{alpha} E_{i}), where the edge set of G is partitioned into alpha color classes, is called an alphaedgecolored graph. A natural extension of the FDELETION problem to edgecolored graphs is the alphaSIMULTANEOUS FDELETION problem. In the latter problem, we are given an alphaedgecolored graph G and the goal is to find a set S of at most k vertices such that each graph G_i\S, where G_i = (V, E_i) and 1 <= i <= alpha, is in F. In this work, we study alphaSIMULTANEOUS FDELETION for F being the family of forests. In other words, we focus on the alphaSIMULTANEOUS FEEDBACK VERTEX SET (alphaSIMFVS) problem. Algorithmically, we show that, like its classical counterpart, alphaSIMFVS parameterized by k is fixedparameter tractable (FPT) and admits a polynomial kernel, for any fixed constant alpha. In particular, we give an algorithm running in 2^{O(alpha * k)} * n^{O(1)} time and a kernel with O(alpha * k^{3(alpha + 1)}) vertices. The running time of our algorithm implies that alphaSIMFVS is FPT even when alpha in o(log(n)). We complement this positive result by showing that for alpha in O(log(n)), where n is the number of vertices in the input graph, alphaSIMFVS becomes W[1]hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).
BibTeX  Entry
@InProceedings{agrawal_et_al:LIPIcs:2016:5708,
author = {Akanksha Agrawal and Daniel Lokshtanov and Amer E. Mouawad and Saket Saurabh},
title = {{Simultaneous Feedback Vertex Set: A Parameterized Perspective}},
booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages = {7:17:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770019},
ISSN = {18688969},
year = {2016},
volume = {47},
editor = {Nicolas Ollinger and Heribert Vollmer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5708},
URN = {urn:nbn:de:0030drops57084},
doi = {10.4230/LIPIcs.STACS.2016.7},
annote = {Keywords: parameterized complexity ,feedback vertex set, kernel, edgecolored graphs}
}
Keywords: 

parameterized complexity ,feedback vertex set, kernel, edgecolored graphs 
Collection: 

33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) 
Issue Date: 

2016 
Date of publication: 

16.02.2016 