  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.
