β-Stars or On Extending a Drawing of a Connected Subgraph

Abstract

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 Γ_H of an induced connected subgraph H can be extended with at most min{h/2, β+log_2(h)+1} bends per edge, where β is the largest star complexity of a face of Γ_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 [12].