Hardness and Parameterized Algorithms on Rainbow Connectivity problem

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such that G is (strongly) rainbow connected. In this paper we study the rainbow connectivity problem and the strong rainbow connectivity problem from a computational point of view. Our main results can be summarised as below: • For every fixed k ≥ 3, it is NP-Complete to decide whether src(G) ≤ k even when the graph G is bipartite. • For every fixed odd k ≥ 3, it is NP-Complete to decide whether rc(G) ≤ k. This resolves one of the open problems posed by Chakraborty et al. (J. Comb. Opt., 2011) where they prove the hardness for the even case. • The following problem is fixed parameter tractable: Given a graph G, determine the maximum number of pairs of vertices that can be rainbow connected using two colors. • For a directed graph G, it is NP-Complete to decide whether rc(G) ≤ 2.