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

  • Tamara Mchedlidze
  • Jérôme UrhausenEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

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 [12].

Notes

Acknowledgment

The authors thank Martin Nöllenburg and Ignaz Rutter for the discussions of this problem back in 2012. Jérôme Urhausen was supported by the Netherlands Organisation for Scientific Research under project 612.001.651.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Utrecht UniversityUtrechtThe Netherlands

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