Inserting one edge into a simple drawing is hard

Arroyo Guevara AM, Klute F, Parada I, Seidel R, Vogtenhuber B, Wiedera T. 2020. Inserting one edge into a simple drawing is hard. Graph-Theoretic Concepts in Computer Science. WG: Workshop on Graph-Theoretic Concepts in Computer Science, LNCS, vol. 12301, 325–338.

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Author
Arroyo Guevara, Alan MIST Austria; Klute, Fabian; Parada, Irene; Seidel, Raimund; Vogtenhuber, Birgit; Wiedera, Tilo
Department
Series Title
LNCS
Abstract
A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ , it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles.
Publishing Year
Date Published
2020-10-09
Proceedings Title
Graph-Theoretic Concepts in Computer Science
Volume
12301
Page
325-338
Conference
WG: Workshop on Graph-Theoretic Concepts in Computer Science
Conference Location
Leeds, United Kingdom
Conference Date
2020-06-24 – 2020-06-26
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Arroyo Guevara AM, Klute F, Parada I, Seidel R, Vogtenhuber B, Wiedera T. Inserting one edge into a simple drawing is hard. In: Graph-Theoretic Concepts in Computer Science. Vol 12301. Springer Nature; 2020:325-338. doi:10.1007/978-3-030-60440-0_26
Arroyo Guevara, A. M., Klute, F., Parada, I., Seidel, R., Vogtenhuber, B., & Wiedera, T. (2020). Inserting one edge into a simple drawing is hard. In Graph-Theoretic Concepts in Computer Science (Vol. 12301, pp. 325–338). Leeds, United Kingdom: Springer Nature. https://doi.org/10.1007/978-3-030-60440-0_26
Arroyo Guevara, Alan M, Fabian Klute, Irene Parada, Raimund Seidel, Birgit Vogtenhuber, and Tilo Wiedera. “Inserting One Edge into a Simple Drawing Is Hard.” In Graph-Theoretic Concepts in Computer Science, 12301:325–38. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-60440-0_26.
A. M. Arroyo Guevara, F. Klute, I. Parada, R. Seidel, B. Vogtenhuber, and T. Wiedera, “Inserting one edge into a simple drawing is hard,” in Graph-Theoretic Concepts in Computer Science, Leeds, United Kingdom, 2020, vol. 12301, pp. 325–338.
Arroyo Guevara AM, Klute F, Parada I, Seidel R, Vogtenhuber B, Wiedera T. 2020. Inserting one edge into a simple drawing is hard. Graph-Theoretic Concepts in Computer Science. WG: Workshop on Graph-Theoretic Concepts in Computer Science, LNCS, vol. 12301, 325–338.
Arroyo Guevara, Alan M., et al. “Inserting One Edge into a Simple Drawing Is Hard.” Graph-Theoretic Concepts in Computer Science, vol. 12301, Springer Nature, 2020, pp. 325–38, doi:10.1007/978-3-030-60440-0_26.

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