## Abstract

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of geometry, combinatorics and topology. We define a \({\mathbb {Z}}_2\) valued *self-intersection invariant* (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments accessible to mathematicians not specialized in any of the areas discussed. So most part of this survey could be studied before textbooks on algebraic topology, as an introduction to starting ideas of algebraic topology motivated by algorithmic, combinatorial and geometric problems.

This is a preview of subscription content, log in to check access.

## Notes

- 1.
Examples are definition of the mapping degree [Matoušek 2008, Sect. 2.4], [Skopenkov 2020, Sect. 8] and definition of the Hopf invariant via linking, i.e., via intersection [Skopenkov 2020, Sect. 8]. Importantly, ‘secondary’ not only ‘primary’ invariants allow interpretations in terms of

*framed*intersections; for a recent application see [Skopenkov 2017a]. - 2.
The ‘minimal generality’ principle (to introduce important ideas in non-technical particular cases) was put forward by classical figures in mathematics and mathematical exposition, in particular by V. Arnold. Cf. ‘detopologization’ tradition described in [Matoušek et al. 2012, Historical notes in Sect. 1].

- 3.
The common term for this notion is

*a graph without loops and multiple edges*or*a simple graph*. - 4.
- 5.
See proof in [Skopenkov 2018c, Sect. 1.6]. Proposition 1.1.2 and [Skopenkov 2018c, 1.6.1] are not formally used in this paper. However, they illustrate by two-dimensional examples how boolean functions appear in the study of embeddings. This is one of the ideas behind recent higher-dimensional

*NP*-hardness Theorem 3.2.3.b. - 6.
We do not require that ‘no isolated vertex lies on any of the segments’ because this property can always be achieved.

- 7.
Rigorous definition of the notion of algorithm is complicated, so we do not give it here. Intuitive understanding of algorithms is sufficient to read this text. To be more precise, the above statement means that there is an algorithm for calculating the function from the set of all graphs to \(\{0,1\}\), which maps graph to 1 if the graph is linearly realizable in the plane, and to 0 otherwise. All other statements on algorithms in this paper can be formalized analogously.

- 8.
Then any two of the polygonal lines either are disjoint or intersect by a common end vertex. We do not require that ‘no isolated vertex lies on any of the polygonal lines’ because this property can always be achieved. See an equivalent definition of planarity in the beginning of Sect. 1.4.

- 9.
Since for a planar graph with

*n*vertices and*e*edges we have \(e \le 3n-6\) and since there are planar graphs with*n*vertices and*e*edges such that \(e=3n-6\), the ‘complexity’ in the number of edges is ‘the same’ as the ‘complexity’ in the number of vertices. - 10.
- 11.
The number \(L\cdot P\) is defined in Sect. 1.5.4.

This version of the Stokes theorem shows that the complement to

*L*has a*Möbius–Alexander numbering*, i.e. a ‘chess-board coloring by integers’ (so that the colors of the adjacent domains are different by \(\pm 1\) depending on the orientations; the ends of a polygonal line*P*have the same color if and only if \(L\cdot P=0\)).See more in [https://en.wikipedia.org/wiki/Winding_number].

- 12.
This is an elementary interpretation in the spirit of [Schöneborn 2004, Schöneborn and Ziegler 2005] of the

*r*-tuple algebraic intersection number \(fD^{n_1}\ldots fD^{n_r}\) of a general position map \(f:D^{n_1}\sqcup \cdots \sqcup D^{n_r}\rightarrow {\mathbb {R}}^2\), where \(n_1,\ldots ,n_r\subset \{0,1,2\}\) and \(n_1+\cdots +n_r=2r-2\) [Mabillard and Wagner 2015, Sect. 2.2]. This agrees with [Mabillard and Wagner 2015, Sect. 2.2] by [Mabillard and Wagner 2015, Lemma 27.b]. For a degree interpretation see [Skopenkov 2018c, Assertion 2.5.4]. - 13.
This is the \(d(r-1)\)-skeleton of the

*simplicial**r*-*fold deleted product*of*K*. Cf. [Skopenkov 2018a, Sect. 1.4]. - 14.
This agrees up to sign with the definition of [Mabillard and Wagner 2015, Lemma 41.b] because by [Mabillard and Wagner 2015, (13) in p. 17] \(\varepsilon _{2,2,\ldots ,2,0}\) is even and \(\varepsilon _{2,2,\ldots ,2,1,1}\) is odd.

The

*r*-fold intersection cocycle depends on an arbitrary choice of orientations, but the triviality condition defined below does not. - 15.
Here

*NP*-hardness means that using a devise which solves this problem EMBED(k,d) at 1 step, we can construct an algorithm which is polynomial in*n*and which recognizes if a boolean function of*n*variables is identical zero, the function given as a disjunction of some conjunctions of variables or their negations (e.g. \(f(x_1,x_2,x_3,x_4)=x_1x_2{\overline{x}}_3\vee {\overline{x}}_2x_3x_4\vee {\overline{x}}_1x_2x_4\)). M. Tancer suggests that it is plausible to approach the conjecture the same way as in [Matoušek et al. 2011, Skopenkov and Tancer 2017]. Namely, one can possibly triangulate the gadgets in advance and glue them together so that the ‘embeddable gadgets’ would be linearly embeddable with respect to the prescribed triangulations. By using the same triangulation on gadgets of same type, one can achieve polynomial size triangulation. Realization of this idea should be non-trivial. - 16.

## References

Arnold, V.I.: Topological invariants of plane curves and caustics, University Lecture Series, vol. 5. American Mathematical Society, Providence (1995)

Avvakumov, S., Karasev, R.: Envy-free division using mapping degree (2019). arXiv:1907.11183

Avvakumov, S., Karasev, R., Skopenkov, A.: Stronger counterexamples to the topological Tverberg conjecture. (2019a). arxiv:1908.08731 (

**submitted**)Avvakumov S, Mabillard, I., Skopenkov, A., Wagner, U.: Eliminating higher-multiplicity intersections, III. Codimension 2, Israel J. Math. (to appear). (2019b). arxiv:1511.03501

Bajmóczy, E.G., Bárány, I.: On a common generalization of Borsuk’s and Radon’s theorem. Acta Math. Acad. Sci. Hungar.

**34**(3), 347–350 (1979)Bárány, I., Shlosman, S.B., Szűcs, A.: On a topological generalization of a theorem of Tverberg. J. Lond. Math. Soc. (II. Ser.)

**23**, 158–164 (1981)Bárány, I., Blagojević, P.V.M., Ziegler, G.M.: Tverberg’s theorem at 50: extensions and counterexamples. Not. AMS

**63**(7), 732–739 (2016)Blagojevič, P.V.M., Ziegler, G.M.: Beyond the Borsuk–Ulam theorem: the topological Tverberg story. In: Loebl, M., Nešetřil, J., Thomas, R. (eds.) A Journey Through Discrete Mathematics, pp. 273–341, (2016). arXiv:1605.07321v2

Blagojević, P.V.M., Frick, F., Ziegler, G.M.: Tverberg plus constraints. Bull. Lond. Math. Soc.

**46**(5), 953–967 (2014). arXiv: 1401.0690Blagojević, P.V.M., Matschke, B., Ziegler, G.M.: Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc.

**17**(4), 739–754 (2015). arXiv:0910.4987Boyer, J.M., Myrvold, W.J.: On the cutting edge: simplified \(O(n)\) planarity by edge addition. J. Graph Alg. Appl.

**8**(3), 241–273 (2004)Čadek, M., Krčál, M., Vokřínek, L.: Algorithmic solvability of the lifting-extension problem (2019). arXiv:1307.6444

Chernov, A., Daynyak, A., Glibichuk, A., Ilyinskiy, M., Kupavskiy, A., Raigorodskiy, A., Skopenkov, A.: Elements of Discrete Mathematics As a Sequence of Problems (in Russian), MCCME, Moscow (2016). http://www.mccme.ru/circles/oim/discrbook.pdf. Accessed 7 Nov 2019

de Mesmay, A., Rieck, Y., Sedgwick, E., Tancer, M.: Embeddability in \({\mathbb{R}}^{3}\) is NP-hard (2019). arXiv:1708.07734

Enne, A., Ryabichev, A., Skopenkov, A., Zaitsev, T.: Invariants of graph drawings in the plane (2019). http://www.turgor.ru/lktg/2017/6/index.htm. Accessed 7 Nov 2019

Filakovsky, M., Wagner, U., Zhechev, S.: Embeddability of simplicial complexes is undecidable. Oberwolfach reports (2019)

**(to appear)**Fokkink, R.: A forgotten mathematician. Eur. Math. Soc. Newsl.

**52**, 9–14 (2004)Freedman, M.H., Krushkal, V.S., Teichner, P.: Van Kampen’s embedding obstruction is incomplete for 2-complexes in \({{\mathbb{R}}^{4}}\). Math. Res. Lett.

**1**, 167–176 (1994)Frick, F.: Counterexamples to the topological Tverberg conjecture, Oberwolfach reports (2015). arXiv:1502.00947

Frick, F.: On affine Tverberg-type results without continuous generalization (2017). arXiv:1702.05466

Gromov, M.: Singularities, expanders and topology of maps. Part 2: from combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal.

**20**(2), 416–526 (2010)Gross, J.L., Rosen, R.H.: A linear time planarity algorithm for 2-complexes. J. ACM

**26**(4), 611–617 (1979)Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. Assoc. Comput. Mach.

**21**(4), 549–568 (1974)Khoroshavkina, N.: A simple characterization of graphs of cutwidth (2019). arXiv:1811.06716

Lin, Y., Yang, A.: On 3-cutwidth critical graphs. Discret. Math.

**275**, 339–346 (2004)Lovasz, L., Schrijver, A.: A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs. Proc. AMS

**126**(5), 1275–1285 (1998)Mabillard, I., Wagner, U.: Eliminating Higher-Multiplicity Intersections, I. A Whitney trick for Tverberg-Type problems (2015). arXiv:1508.02349

Mabillard, I., Wagner, U.: Eliminating Higher-Multiplicity Intersections, II. The deleted product criterion in the \(r\)-metastable range (2016). arXiv:1601.00876

Matoušek, J.: Using the Borsuk-Ulam theorem: lectures on topological methods in combinatorics and geometry. Springer, Berlin (2008)

Matoušek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in \({\mathbb{R}}^{d}\). J. Eur. Math. Soc.

**13**(2), 259–295 (2011). arXiv:0807.0336Matoušek, J., Tancer, M., Wagner, U.: A geometric proof of the colored Tverberg theorem. Discret. Comp. Geom.

**47**(2), 245–265 (2012). arXiv:1008.5275Matoušek, J., Sedgwick, E., Tancer, M., Wagner, U.: Embeddability in the 3-sphere is decidable. J. ACM

**65**(1), 1–49 (2018). arXiv:1402.0815Özaydin, M.: Equivariant maps for the symmetric group, unpublished (2019). http://minds.wisconsin.edu/handle/1793/63829. Accessed 7 Nov 2019

Sarkaria, K.S.: A generalized Van Kampen-Flores theorem. Proc. Am. Math. Soc.

**111**, 559–565 (1991)Schöneborn, T.: On the topological Tverberg theorem (2004). arXiv:math/0405393

Schöneborn, T., Ziegler, G.: The topological Tverberg Theorem and winding numbers. J. Combin. Theory Ser. A

**112**(1), 82–104 (2005). arXiv:math/0409081Shlosman, S.: Topological Tverberg theorem: the proofs and the counterexamples. Russ. Math. Surv.

**73**(2), 175–182 (2018). arXiv:1804.03120Skopenkov, M.: Embedding products of graphs into Euclidean spaces. Fund. Math.

**179**, 191–198 (2003). arXiv:0808.1199Skopenkov, A.: A new invariant and parametric connected sum of embeddings. Fund. Math.

**197**, 253–269 (2007). arXiv:math/0509621Skopenkov, A.: Embedding and knotting of manifolds in Euclidean spaces. Lond. Math. Soc. Lect. Notes

**347**, 248–342 (2008). arXiv:math/0604045Skopenkov, A.: Realizability of hypergraphs and Ramsey link theory. (2014). arxiv:1402.0658

Skopenkov, A.: Eliminating higher-multiplicity intersections in the metastable dimension range (2017a). arxiv:1704.00143

Skopenkov, A.: On the Metastable Mabillard–Wagner Conjecture (2017b). arxiv:1702.04259

Skopenkov, A.: A user’s guide to the topological Tverberg Conjecture, Russian Math. Surveys, 73:2 (2018a), 323–353. Earlier version: arXiv:1605.05141v4. §4 available as A. Skopenkov, On van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems. arXiv:1704.00300

Skopenkov, A.: Stability of intersections of graphs in the plane and the van Kampen obstruction. Topol. Appl.

**240**, 259–269 (2018b). arXiv:1609.03727Skopenkov, A.: Invariants of graph drawings in the plane. Full author’s version (2018c). arXiv:1805.10237

Skopenkov, A.: Algebraic topology from algorithmic viewpoint, draft of a book, mostly in Russian (2019). http://www.mccme.ru/circles/oim/algor.pdf. Accessed 7 Nov 2019

Skopenkov, A.: Algebraic topology from geometric viewpoint (in Russian), MCCME, Moscow, 2nd edition (2020). http://www.mccme.ru/circles/oim/obstruct.pdf. Accessed 7 Nov 2019

Skopenkov, A., Tancer, M.: Hardness of Almost Embedding Simplicial Complexes in \(R^{d}\). Discret. Comp. Geom. (to appear) (2017). arXiv:1703.06305

Tamassia, R. (Ed.): Handbook of Graph Drawing and Visualization. Chapman and Hall/CRC (2019)

Thilikos, D.M., Serna, M., Bodlaender, H.L.: Cutwidth I: a linear time fixed parameter algorithm. J. Algorithms

**56**(1), 1–24 (2005)Ummel, B.: The product of nonplanar complexes does not imbed in 4-space. Trans. Am. Math. Soc.

**242**, 319–328 (1978)van Kampen, E.R.: Remark on the address of S. S. Cairns, in Lectures in Topology, pp. 311–313. University of Michigan Press, Ann Arbor (1941)

Volovikov, AYu.: On the van Kampen-Flores theorem. Math. Notes

**59**(5), 477–481 (1996a)Volovikov, AYu.: On a topological generalization of the Tverberg theorem. Math. Notes

**59**(3), 324–326 (1996b)Vučić, A., Živaljević, R.T.: Note on a conjecture of Sierksma. Discret. Comput. Geom.

**9**, 339–349 (1993)Ziegler, G.M.: 3N colored points in a plane. Not. AMS

**58**(4), 550–557 (2011)

## Author information

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported in part by the Russian Foundation for Basic Research Grant No. 19-01-00169 and by Simons-IUM Fellowship.

## Rights and permissions

## About this article

### Cite this article

Skopenkov, A. Invariants of Graph Drawings in the Plane.
*Arnold Math J.* (2020). https://doi.org/10.1007/s40598-019-00128-5

Received:

Revised:

Accepted:

Published: