## Abstract

For a family of graphs F, an n-vertex 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 (multi) graph G = (V, ?^{a}_{i}_{=1}E_{i}), where the edge set of G is partitioned into a color classes, is called an a-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_{1}, . . ., F_{a})-Deletion problem. In the latter problem, we are given an a-edge-colored 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 = a, is in F_{i}. Recently, a subset of the authors considered the aforementioned problem with F_{1} = . . . = F_{a} being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and a and can be solved in O(2^{O}(ak^{)}) time^{1}. In this work, we initiate the investigation of the complexity of Simultaneous (F_{1}, . . ., F_{a})-Deletion with di erent families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_{i}s is the class of bipartite graphs and the rest (if any) are forests. We show that if F_{1} is the family of all bipartite graphs and each of F_{2} = F_{3} = . . . = F_{a} is the family of all forests then the problem is fixed-parameter tractable parameterized by k and a. However, even when F_{1} and F_{2} are both the family of all bipartite graphs, then the Simultaneous (F_{1}, F_{2})-Deletion problem itself is already W[1]-hard.

Original language | English |
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Title of host publication | 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017 |

Editors | Satya Lokam, R. Ramanujam |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770552 |

DOIs | |

State | Published - 1 Jan 2018 |

Externally published | Yes |

Event | 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017 - Kanpur, India Duration: 12 Dec 2017 → 14 Dec 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 93 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017 |
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Country/Territory | India |

City | Kanpur |

Period | 12/12/17 → 14/12/17 |

## Keywords

- Edge-colored graphs
- Feedback vertex set
- Odd cycle transversal
- Parameterized complexity
- Simultaneous deletion

## ASJC Scopus subject areas

- Software