In this paper, we propose a generic concurrent directed graph (for shared memory architecture) that is concurrently being updated by threads adding/deleting vertices and edges. The graph is constructed by the composition of the well known concurrent list-based set data-structure from the literature. Our construction is generic, in the sense that it can be used to obtain various progress guarantees, depending on the granularity of the underlying concurrent set implementation - either blocking or non-blocking. We prove that the proposed construction is linearizable by identifying its linearization points. Finally, we compare the performance of all the variants of the concurrent graph data-structure along with its sequential implementation. We observe that our concurrent graph data-structure mimics the performance of the concurrent list based set.