# SIMULTANEOUS Feedback Vertex set: A parameterized perspective

Akanksha Agrawal, Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

## Abstract

For a family of graphs F, a graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex SET, and ODD CYCLE TRANSVERSAL. A graph G = (V,∪αi=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colored graph. A natural extension of the F-DELETION problem to edge-colored graphs is the α-SIMULTANEOUS F-DELETION problem. In the latter problem, we are given an α-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph Gi \ S, where Gi = (V, Ei) and 1 ≤ i ≤ α, is in F. In this work, we study α-SIMULTANEOUS F-DELETION for F being the family of forests. In other words, we focus on the α-SIMULTANEOUS FEEDBACK VERTEX SET (α-SIMFVS) problem. Algorithmically, we show that, like its classical counterpart, α-SIMFVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant α. In particular, we give an algorithm running in 2O(αk)nO(1) time and a kernel with O(αk3(α+1)) vertices. The running time of our algorithm implies that α-SimFVS is FPT even when α ∈ o(log n). We complement this positive result by showing that for α ∈ O(log n), where n is the number of vertices in the input graph, α-SIMFVS becomes W-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).

Original language English 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016 Heribert Vollmer, Nicolas Ollinger Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959770019 https://doi.org/10.4230/LIPIcs.STACS.2016.7 Published - 1 Feb 2016 Yes 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016 - Orleans, FranceDuration: 17 Feb 2016 → 20 Feb 2016

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 47 1868-8969

### Conference

Conference 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016 France Orleans 17/02/16 → 20/02/16

## Keywords

• Edge-colored graphs
• Feedback Vertex SET
• Kernel
• Parameterized complexity

• Software

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