Untangling a planar graph

Authors:
Andreas Spillner;Alexander Wolff
Affiliations:
School of Computing Sciences, University of East Anglia, Norwich, UK;Faculteit Wiskunde en Informatica, TU Eindhoven, The Netherlands
Venue:
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
Year:
2008

Citing 4
Cited 2

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Efficient Planarity Testing

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GD'07 Proceedings of the 15th international conference on Graph drawing
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WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science

Note: On the obfuscation complexity of planar graphs

Theoretical Computer Science
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Discrete Applied Mathematics

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Abstract

In John Tantalo's on-line game Planarity the player is given a non-plane straight-line drawing of a planar graph. The aim is to make the drawing plane as quickly as possible by moving vertices. Pach and Tardos have posed a related problem: can any straight-line drawing of any planar graph with n vertices be made plane by vertex moves while keeping Ω(nƐ) vertices fixed for some absolute constant Ɛ 0? It is known that three vertices can always be kept (if n ≥ 5). We still do not solve the problem of Pach and Tardos, but we report some progress. We prove that the number of vertices that can be kept actually grows with the size of the graph. More specifically, we give a lower bound of Ω(√log n/ log log n) on this number. By the same technique we show that in the case of outerplanar graphs we can keep a lot more, namely Ω(√n) vertices. We also construct a family of outerplanar graphs for which this bound is asymptotically tight.