Abstract
This paper considers 1-string representations of planar graphs that are order-preserving in the sense that the order of crossings along the curve representing vertex v is the same as the order of edges in the clockwise order around v in the planar embedding. We show that this does not exist for all planar graphs (not even for all planar 3-trees), but show existence for some subclasses of planar partial 3-trees. In particular, for outer-planar graphs it can be order-preserving and outer-string in the sense that all ends of strings are on the outside of the representation.
T. Biedl—Research supported by NSERC.
M. Derka—was supported by the Vanier CGS.
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Notes
- 1.
One could distinguish this further by whether both ends must be on the contour or whether one end suffices. All our outer-string constructions have both ends on the contour, while all our impossibility-results hold even if only one end is required to be on the contour, so the distinction does not matter for the results in our paper.
- 2.
Once we fix how to break up the cyclic order at all vertices, there is a construction that describes the order-preserving 1-string representation as a graph H and so that it can be realized if and only if H is planar. Hence the problem is interesting only if we keep this choice.
- 3.
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Biedl, T., Derka, M. (2017). Order-Preserving 1-String Representations of Planar Graphs. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_22
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