  A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected
graph G is a mapping $\lambda$ from the set of vertices (resp. the set of
edges) of G to the set of labels {1,. .. , r} such that no non-trivial
automorphism of G preserves all the vertex (resp. edge) labels. The
distinguishing number D(G) and the distinguishing index D (G) of G are then the
smallest r for which G admits a distinguishing r-vertex-labelling or
r-edge-labelling, respectively. The distinguishing chromatic number D $\chi$
(G) and the distinguishing chromatic index D $\chi$ (G) are defined similarly,
with the additional requirement that the corresponding labelling must be a
proper colouring. These notions readily extend to oriented graphs, by
considering arcs instead of edges. In this paper, we study the four
corresponding parameters for oriented graphs whose underlying graph is a path,
a cycle, a complete graph or a bipartite complete graph. In each case, we
determine their minimum and maximum value, taken over all possible orientations
of the corresponding underlying graph, except for the minimum values for
unbalanced complete bipartite graphs K m,n with m = 2, 3 or 4 and n > 3, 6 or
13, respectively, or m $\ge$ 5 and n > 2 m -- m 2 , for which we only provide
upper bounds.
