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Invariants of Graph Drawings in the Plane

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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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  1. 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. 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. 3.

    The common term for this notion is a graph without loops and multiple edges or a simple graph.

  4. 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. 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. 6.

    We do not require that ‘no isolated vertex lies on any of the segments’ because this property can always be achieved.

  7. 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. 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. 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. 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. 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. 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. 13.

    This is the \(d(r-1)\)-skeleton of the simplicial  r-fold deleted product of K. Cf. [Skopenkov 2018a, Sect. 1.4].

  14. 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. 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. 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.


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

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