Inserting an Edge into a Geometric Embedding
The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e [10], is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs. Unfortunately, some graphs do not have a geometric embedding \documentclass[12pt...
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Published in | Graph Drawing and Network Visualization pp. 402 - 415 |
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Main Authors | , |
Format | Book Chapter |
Language | English |
Published |
Cham
Springer International Publishing
|
Series | Lecture Notes in Computer Science |
Online Access | Get full text |
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Summary: | The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e [10], is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs. Unfortunately, some graphs do not have a geometric embedding \documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} such that \documentclass[12pt]{minimal}
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\begin{document}$$\varGamma +e$$\end{document} has the same number of crossings as the embedding \documentclass[12pt]{minimal}
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\begin{document}$$G+e$$\end{document}. This motivates the study of the computational complexity of the following problem: Given a combinatorially embedded graph G, compute a geometric embedding \documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} that has the same combinatorial embedding as G and that minimizes the crossings of \documentclass[12pt]{minimal}
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\begin{document}$$\varGamma +e$$\end{document}. We give polynomial-time algorithms for special cases and prove that the general problem is fixed-parameter tractable in the number of crossings. Moreover, we show how to approximate the number of crossings by a factor \documentclass[12pt]{minimal}
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\begin{document}$$(\varDelta -2)$$\end{document}, where \documentclass[12pt]{minimal}
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\begin{document}$$\varDelta $$\end{document} is the maximum vertex degree of G. |
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Bibliography: | Work was partially supported by grant WA 654/21-1 of the German Research Foundation (DFG). |
ISBN: | 9783030044138 3030044130 |
ISSN: | 0302-9743 1611-3349 |
DOI: | 10.1007/978-3-030-04414-5_29 |