Generalized rainbow connectivity of graphs

Let C={c₁, c₂,..., c_k} be a set of k colors, and let ℓ→=(ℓ1,ℓ2,...,ℓk) be a k-tuple of nonnegative integers ℓ₁, ℓ₂,..., 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}.