A directed multigraph is said vulnerable if it can generate Braess paradox in Traffic Networks. In this paper, we give a graph-theoretic characterisation of vulnerable directed multigraphs; analogous results appeared in the literature only for undirected multigraphs and for a specific family of directed multigraphs. The proof of our characterisation also provides an algorithm that checks if a multigraph is vulnerable in O(|V| |E|^2); this is the first polynomial time algorithm that checks vulnerability for general directed multigraphs. The resulting algorithm also contributes to another well known problem, i.e. the directed subgraph homeomorphism problem without node mapping, by providing another pattern graph for which a polynomial time algorithm exists.