Generalized Rainbow Connectivity of Graphs

Abstract

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 }.

This work is partially supported by JSPS KAKENHI Grant Numbers 22700001, 23500001 and 23700003.