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 ΓH of an induced connected subgraph H can be extended with at
most min{h/2,β+log2(h)+1} bends per edge, where β is the
largest star complexity of a face of ΓH 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].Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018