# Inserting an Edge into a Geometric Embedding

• Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

## Abstract

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 $$\varGamma$$ such that $$\varGamma +e$$ has the same number of crossings as the embedding $$G+e$$. This motivates the study of the computational complexity of the following problem: Given a combinatorially embedded graph G, compute a geometric embedding $$\varGamma$$ that has the same combinatorial embedding as G and that minimizes the crossings of $$\varGamma +e$$. 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 $$(\varDelta -2)$$, where $$\varDelta$$ is the maximum vertex degree of G.

## References

1. 1.
Chimani, M., Gutwenger, C., Mutzel, P., Wolf, C.: Inserting a vertex into a planar graph. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 375–383 (2009)
2. 2.
Chimani, M., Hlinený, P.: Inserting multiple edges into a planar graph. In: Fekete,S., Lubiw, A. (eds.) Proceedings of the 32nd Annual Symposium on Computational Geometry, SoCG 2016. Leibniz International Proceedings in Informatics (LIPIcs), vol. 51, pp. 30:1–30:15. Schloss DagstuhlLeibniz-Zentrum fuer Informatik (2016).
3. 3.
Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2015).
4. 4.
Eades, P., Hong, S.H., Liotta, G., Katoh, N., Poon, S.H.: Straight-line drawability of a planar graph plus an edge. In: Dehne, F., Sack, J.R., Stege, U. (eds.) Proceedings of the 14th International Symposium on Algorithms and Data Structures, WADS 2015, pp. 301–313 (2015). Google Scholar
5. 5.
Eilam-Tzoreff, T.: The disjoint shortest paths problem. Discrete Appl. Math. 85(2), 113–138 (1998).
6. 6.
Even, S., Itai, A., Shamir, A.: On the complexity of time table and multi-commodity flow problems. In: 16th Annual Symposium on Foundations of Computer Science, SFCS 1975, pp. 184–193, October 1975.
7. 7.
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theory Comput. Syst. 10(2), 111–121 (1980).
8. 8.
Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)
9. 9.
Gutwenger, C., Klein, K., Mutzel, P.: Planarity testing and optimal edge insertion with embedding constraints. J. Graph Algorithms Appl. 12(1), 73–95 (2008).
10. 10.
Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41(4), 289–308 (2005).
11. 11.
Kobayashi, Y., Sommer, C.: On shortest disjoint paths in planar graphs. Discrete Optim. 7(4), 234–245 (2010).
12. 12.
Kobourov, S.G.: Force-directed drawing algorithms. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, pp. 383–408. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
13. 13.
Radermacher, M., Reichard, K., Rutter, I., Wagner, D.: A geometric heuristic for rectilinear crossing minimization. In: Pagh, R., Venkatasubramanian, S. (eds.) Proceedings of the 20th Workshop on Algorithm Engineering and Experiments, ALENEX 2018, pp. 129–138 (2018).
14. 14.
Radermacher, M., Rutter, I.: Inserting an edge into a geometric embedding. ArXiv e-prints (2018). https://arxiv.org/abs/1807.11711v1