Generalized Rainbow Connectivity of Graphs

Let C = {c 1, c 2, …, c k } be a set of k colors, and let l = (l1, l2, …, l k ) 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 such that the number of edges in 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 characterize the computational complexity of the problem with regards 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 }.