Arc-distinguishing of orientations of graphs

A distinguishing index of a (di) graph is the minimum number of colours in an edge (or arc) colouring such that the identity is the only automorphism that preserves that colouring. We investigate the minimum and maximum value of the distinguishing index over all orientations of a given graph G. We present sharp results for these parameters in terms of the distinguishing index of G for trees, unbalanced bipartite graphs, traceable graphs and claw-free graphs. With this, we answer the question of Meslem and Sopena.