  We consider the problem of extending the drawing of a subgraph of a given
plane graph to a drawing of the entire graph using straight-line and polyline
edges. We define the notion of star complexity of a polygon and show that a
drawing $\Gamma_H$ of an induced connected subgraph $H$ can be extended with at
most $\min\{ h/2, \beta + \log_2(h) + 1\}$ bends per edge, where $\beta$ is the
largest star complexity of a face of $\Gamma_H$ and $h$ is the size of the
largest face of $H$. This result significantly improves the previously known
upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show
that our bound is worst case optimal up to a small additive constant.
Additionally, we provide an indication of complexity of the problem of testing
whether a star-shaped inner face can be extended to a straight-line drawing of
the graph; this is in contrast to the fact that the same problem is solvable in
linear time for the case of star-shaped outer face [9] and convex inner face
[13].
