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A path in an edge colored graph is said to be a rainbow path if no two edgeson the path have the same color. An edge colored graph is (strongly) rainbowconnected if there exists a (geodesic) rainbow path between every pair ofvertices. 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 thegraph such that G is (strongly) rainbow connected. In this paper we study therainbow connectivity problem and the strong rainbow connectivity problem from acomputational point of view. Our main results can be summarised as below:  1) For every fixed k >= 3, it is NP-Complete to decide whether src(G) <= keven when the graph G is bipartite.  2) 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.  3) The following problem is fixed parameter tractable: Given a graph G,determine the maximum number of pairs of vertices that can be rainbow connectedusing two colors.  4) For a directed graph G, it is NP-Complete to decide whether rc(G) <= 2.

Hardness and Parameterized Algorithms on Rainbow Connectivity problem