## Abstract

Let C={c_{1}, c_{2},..., c_{k}} be a set of k colors, and let ℓ→=(ℓ1,ℓ2,...,ℓk) be a k-tuple of nonnegative integers ℓ_{1}, ℓ_{2},..., ℓ_{k}. For a graph G=(V, E), let f:E→C be an edge-coloring of G in which two adjacent edges may have the same color. Then, the graph G edge-colored by f is ℓ→-rainbow connected if every two vertices of G have a path P connecting them such that the number of edges on P that are colored with c_{j} is at most ℓ_{j} for each index j∈{1, 2,..., k}. Given a k-tuple ℓ→ and an edge-colored graph, we study the problem of determining whether the edge-colored graph is ℓ→-rainbow connected. In this paper, we first study the computational complexity of the problem with regard to certain graph classes: the problem is NP-complete even for cacti, while is solvable in polynomial time for trees. We then give an FPT algorithm for general graphs when parameterized by both k and ℓ_{max}=max {ℓ_{j}|1≤j≤k}.

Original language | English |
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Pages (from-to) | 35-42 |

Number of pages | 8 |

Journal | Theoretical Computer Science |

Volume | 555 |

Issue number | C |

DOIs | |

Publication status | Published - 2014 |

## Keywords

- Cactus
- Fixed parameter tractability
- Graph algorithm
- Rainbow connectivity
- Tree

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)