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arXiv · 1808.10366

$\beta$-Stars or On Extending a Drawing of a Connected Subgraph

Tamara Mchedlidze·Jérôme Urhausen
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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 $\Gamma_H$ 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].

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Tamara Mchedlidze, Jérôme Urhausen. 2018-08-30. $\beta$-Stars or On Extending a Drawing of a Connected Subgraph. https://arxiv.org/abs/1808.10366

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