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Invariants of Graph Drawings in the Plane

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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.

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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. These are ‘linear’ versions of the nonplanarity of the graphs \(K_5\) and \(K_{3,3}\). But they can be proved easier (because the Parity Lemma 1.3.2.b and [Skopenkov 2020, Intersection Lemma 1.4.4] are not required for the proof).

  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. Example 1.5.4 and Proposition 1.5.6.b explain how \(b_{\sigma ,\tau }\) and \(a_{A,e,\sigma ,\tau }\) naturally appear in the proof.

  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. Analogous problems for maps from graphs to the line are investigated in studies of cutwidth, see [Thilikos et al. 2005, Lin and Yang 2004, Khoroshavkina 2019] and references therein.

References

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

    Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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: 1401.0690

    Article  MathSciNet  MATH  Google Scholar 

  • Blagojević, 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:0910.4987

    Article  MathSciNet  MATH  Google Scholar 

  • Boyer, 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Gross, J.L., Rosen, R.H.: A linear time planarity algorithm for 2-complexes. J. ACM 26(4), 611–617 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  • Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. Assoc. Comput. Mach. 21(4), 549–568 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Book  MATH  Google Scholar 

  • 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:0807.0336

    Article  MathSciNet  MATH  Google Scholar 

  • Matoušek, J., Tancer, M., Wagner, U.: A geometric proof of the colored Tverberg theorem. Discret. Comp. Geom. 47(2), 245–265 (2012). arXiv:1008.5275

    Article  MathSciNet  MATH  Google Scholar 

  • Matoušek, J., Sedgwick, E., Tancer, M., Wagner, U.: Embeddability in the 3-sphere is decidable. J. ACM 65(1), 1–49 (2018). arXiv:1402.0815

    Article  MathSciNet  MATH  Google Scholar 

  • Özaydin, 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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/0409081

    Article  MathSciNet  MATH  Google Scholar 

  • Shlosman, S.: Topological Tverberg theorem: the proofs and the counterexamples. Russ. Math. Surv. 73(2), 175–182 (2018). arXiv:1804.03120

    Article  MathSciNet  MATH  Google Scholar 

  • Skopenkov, M.: Embedding products of graphs into Euclidean spaces. Fund. Math. 179, 191–198 (2003). arXiv:0808.1199

    Article  MathSciNet  MATH  Google Scholar 

  • Skopenkov, A.: A new invariant and parametric connected sum of embeddings. Fund. Math. 197, 253–269 (2007). arXiv:math/0509621

    Article  MathSciNet  MATH  Google Scholar 

  • Skopenkov, A.: Embedding and knotting of manifolds in Euclidean spaces. Lond. Math. Soc. Lect. Notes 347, 248–342 (2008). arXiv:math/0604045

    MathSciNet  MATH  Google Scholar 

  • Skopenkov, 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:1609.03727

    Article  MathSciNet  MATH  Google Scholar 

  • Skopenkov, 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Ummel, B.: The product of nonplanar complexes does not imbed in 4-space. Trans. Am. Math. Soc. 242, 319–328 (1978)

    MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Ziegler, G.M.: 3N colored points in a plane. Not. AMS 58(4), 550–557 (2011)

    MathSciNet  MATH  Google Scholar 

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  1. Moscow Institute of Physics and Technology, Independent University of Moscow, Moscow, Russia

    A. Skopenkov

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

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Skopenkov, A. Invariants of Graph Drawings in the Plane. Arnold Math J. 6, 21–55 (2020). https://doi.org/10.1007/s40598-019-00128-5

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