  The partial representation extension problem generalizes the recognition
problem for classes of graphs defined in terms of vertex representations. We
exhibit circular-arc graphs as the first example of a graph class where the
recognition is polynomially solvable while the representation extension problem
is NP-complete. In this setting, several arcs are predrawn and we ask whether
this partial representation can be completed.
  We complement this hardness argument with tractability results of the
representation extension problem on various subclasses of circular-arc graphs,
most notably on all variants of Helly circular-arc graphs. In particular, we
give linear-time algorithms for extending normal proper Helly and proper Helly
representations. For normal Helly circular-arc representations we give an
$O(n^3)$-time algorithm.
  Surprisingly, for Helly representations, the complexity hinges on the
seemingly irrelevant detail of whether the predrawn arcs have distinct or
non-distinct endpoints: In the former case the previous algorithm can be
extended, whereas the latter case turns out to be NP-complete. We also prove
that representation extension problem of unit circular-arc graphs is
NP-complete.
