# Crossing Minimization in Perturbed Drawings

• Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

## Abstract

Due to data compression or low resolution, nearby vertices and edges of a graph drawing may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map $$\varphi :G\rightarrow \mathbb {R}^2$$. We wish to perturb $$\varphi$$ by an arbitrarily small $$\varepsilon >0$$ into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An $$\varepsilon$$-perturbation, for every $$\varepsilon >0$$, is given by a piecewise linear map $$\psi _\varepsilon :G\rightarrow \mathbb {R}^2$$ with $$\Vert \varphi -\psi _\varepsilon \Vert <\varepsilon$$, where $$\Vert .\Vert$$ is the uniform norm (i.e., $$\sup$$ norm).

We present a polynomial-time solution for this optimization problem when G is a cycle and the map $$\varphi$$ has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and $$\varphi$$ has no spurs, and (ii) when $$\varphi$$ may have spurs and G is a cycle or a union of disjoint paths.

## Keywords

Map approximation C-planarity Crossing number

## References

1. 1.
Fulek, R., Kynčl, J.: Hanani-Tutte for approximating maps of graphs. In: Proceedings of the 34th Symposium on Computational Geometry (SoCG). LIPIcs, vol. 99, pp. 39:1–39:15. Dagstuhl, Germany (2018)Google Scholar
2. 2.
Akitaya, H.A., Fulek, R., Tóth, C.D.: Recognizing weak embeddings of graphs. In: Proceedings of the 29th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 274–292. SIAM (2018)
3. 3.
Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebr. Discret. Methods 4(3), 312–316 (1982)
4. 4.
Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42(5), 1803–1829 (2013)
5. 5.
Akitaya, H.A., Aloupis, G., Erickson, J., Tóth, C.D.: Recognizing weakly simple polygons. Discret. Comput. Geom. 58(4), 785–821 (2017)
6. 6.
Chang, H.C., Erickson, J., Xu, C.: Detecting weakly simple polygons. In: Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1655–1670 (2015)Google Scholar
7. 7.
Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: On embedding a cycle in a plane graph. Discret. Math. 309(7), 1856–1869 (2009)
8. 8.
Hanani, H.: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundam. Math. 23, 135–142 (1934)
9. 9.
Tutte, W.T.: Toward a theory of crossing numbers. J. Comb. Theory 8, 45–53 (1970)
10. 10.
Skopenkov, M.: On approximability by embeddings of cycles in the plane. Topol. Appl. 134(1), 1–22 (2003)
11. 11.
Repovš, D., Skopenkov, A.B.: A deleted product criterion for approximability of maps by embeddings. Topol. Appl. 87(1), 1–19 (1998)
12. 12.
Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F.: Strip planarity testing for embedded planar graphs. Algorithmica 77(4), 1022–1059 (2017)
13. 13.
Jünger, M., Leipert, S., Mutzel, P.: Level planarity testing in linear time. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 224–237. Springer, Heidelberg (1998).
14. 14.
Angelini, P., Lozzo, G.D.: Clustered planarity with pipes. In: Hong, S.H. (ed.) Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC). LIPIcs, vol. 64, pp. 13:1–13:13. Schloss Dagstuhl (2016)Google Scholar
15. 15.
Feng, Q.-W., Cohen, R.F., Eades, P.: How to draw a planar clustered graph. In: Du, D.-Z., Li, M. (eds.) COCOON 1995. LNCS, vol. 959, pp. 21–30. Springer, Heidelberg (1995).
16. 16.
Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995).
17. 17.
Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. J. Graph Algorithms Appl. 9(3), 391–413 (2005)
18. 18.
Hass, J., Scott, P.: Intersections of curves on surfaces. Isr. J. Math. 51(1), 90–120 (1985)