## Abstract

Given a graph G = (V,E), a coloring function χ : V → C, assigning each vertex a color, is called convex if, for every color c ∈ C, the set of vertices with color c induces a connected subgraph of G. In the Convex Recoloring problem a colored graph G_{χ} is given, and the goal is to find a convex coloring χ_{′} of G that recolors a minimum number of vertices. The 2-Convex Recoloring problem (2-CR) is the special case, where the given coloring χ assigns the same color to at most two vertices. 2-CR is known to be NP-hard even if G is a path. We show that weighted 2-CR problem cannot be approximated within any ratio, unless P=NP. On the other hand, we provide an alternative definition of (unweighted) 2-CR in terms of maximum independent set of paths, which leads to a natural greedy algorithm. We prove that its approximation ratio is 3/2 and show that this analysis is tight. This is the first constant factor approximation algorithm for a variant of CR in general graphs. For the special case, where G is a path, the algorithm obtains a ratio of 5/4, an improvement over the previous best known approximation. We also consider the problem of determining whether a given graph has a convex recoloring of size k. We use the above mentioned characterization of 2-CR to show that a problem kernel of size 4k can be obtained in linear time and to design a (Formula presented.) time algorithm for parametrized 2-CR.

Original language | English |
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Title of host publication | Combinatorial Algorithms - 26th International Workshop, IWOCA 2015, Revised Selected Papers |

Editors | William F. Smyth, Zsuzsanna Liptak |

Publisher | Springer Verlag |

Pages | 299-311 |

Number of pages | 13 |

ISBN (Print) | 9783319295152 |

DOIs | |

State | Published - 2016 |

Event | 26th International Workshop on Combinatorial Algorithms, IWOCA 2015 - Verona, Italy Duration: 5 Oct 2015 → 7 Oct 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9538 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 26th International Workshop on Combinatorial Algorithms, IWOCA 2015 |
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Country/Territory | Italy |

City | Verona |

Period | 5/10/15 → 7/10/15 |

### Bibliographical note

Funding Information:D. Rawitz—Supported in part by the Israel Science Foundation (grant no. 497/14).

Publisher Copyright:

© Springer International Publishing Switzerland 2016.