Generalized rainbow connectivity of graphs

Kei Uchizawa, Takanori Aoki, Takehiro Ito, Xiao Zhou

Research output: Contribution to journalArticlepeer-review


Let C={c1, c2,..., ck} be a set of k colors, and let ℓ→=(ℓ1,ℓ2,...,ℓk) be a k-tuple of nonnegative integers ℓ1, ℓ2,..., ℓk. 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 cj is at most ℓj 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 ℓmax=max {ℓj|1≤j≤k}.

Original languageEnglish
Pages (from-to)35-42
Number of pages8
JournalTheoretical Computer Science
Issue numberC
Publication statusPublished - 2014


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


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