Topological graph

This article is about graphs embedded in the plane, with crossings. For graph embeddings on other surfaces, see topological graph theory.

A graph with odd-crossing number 13 and pair-crossing number 15

In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing). A topological graph is also called a drawing of a graph.

An important special class of topological graphs is the class of geometric graphs, where the edges are represented by line segments. (The term geometric graph is sometimes used in a broader, somewhat vague sense.)

The theory of topological graphs is an area of graph theory, mainly concerned with combinatorial properties of topological graphs, in particular, with the crossing patterns of their edges. It is closely related to graph drawing, a field which is more application oriented, and topological graph theory, which focuses on embeddings of graphs in surfaces (that is, drawings without crossings).

Extremal problems for topological graphs

A fundamental problem in extremal graph theory is the following: what is the maximum number of edges that a graph of n vertices can have if it contains no subgraph belonging to a given class of forbidden subgraphs? The prototype of such results is Turán's theorem, where there is one forbidden subgraph: a complete graph with k vertices (k is fixed). Analogous questions can be raised for topological and geometric graphs, with the difference that now certain geometric subconfigurations are forbidden.

Historically, the first instance of such a theorem is due to Paul Erdős, who extended a result of Heinz Hopf and Erika Pannwitz. ^[2] He proved that the maximum number of edges that a geometric graph with n > 2 vertices can have without containing two disjoint edges (that cannot even share an endpoint) is n. John Conway conjectured that this statement can be generalized to simple topological graphs. A topological graph is called "simple" if any pair of its edges share at most one point, which is either an endpoint or a common interior point at which the two edges properly cross. Conway's thrackle conjecture can now be reformulated as follows: A simple topological graph with n > 2 vertices and no two disjoint edges has at most n edges.

The first linear upper bound on the number of edges of such a graph was established by Lovász et al. ^[3] The best known upper bound, 1.3984n, was proved by Fulek and Pach. ^[4] Apart from geometric graphs, Conway's thrackle conjecture is known to be true for x-monotone topological graphs. ^[5] A topological graph is said to be x-monotone if every vertical line intersects every edge in at most one point.

Alon and Erdős ^[6] initiated the investigation of the generalization of the above question to the case where the forbidden configuration consists of k disjoint edges (k > 2). They proved that the number of edges of a geometric graph of n vertices, containing no 3 disjoint edges is O(n). The optimal bound of roughly 2.5n was determined by Černý. ^[7] For larger values of k, the first linear upper bound, ${\displaystyle O(k^{4}n)}$, was established by Pach and Töröcsik. ^[8] This was improved by Tóth to ${\displaystyle O(k^{2}n)}$. ^[9] For the number of edges in a simple topological graph with no k disjoint edges, only an ${\displaystyle O(n\log ^{4k-8}n)}$ upper bound is known. ^[10] This implies that every complete simple topological graph with n vertices has at least ${\displaystyle c{\frac {\log n}{\log \log n}}}$ pairwise disjoint edges, which was improved to ${\displaystyle cn^{{\frac {1}{2}}-\epsilon }}$ by Ruiz-Vargas. ^{[11] [12]} It is possible that this lower bound can be further improved to cn, where c > 0 is a constant.

Quasi-planar graphs

A common interior point of two edges, at which the first edge passes from one side of the second edge to the other, is called a crossing. Two edges of a topological graph cross each other if they determine a crossing. For any integer k > 1, a topological or geometric graph is called k-quasi-planar if it has no k pairwise crossing edges. Using this terminology, if a topological graph is 2-quasi-planar, then it is a planar graph. It follows from Euler's polyhedral formula that every planar graph with n > 2 vertices has at most 3n − 6 edges. Therefore, every 2-quasi-planar graph with n > 2 vertices has at most 3n − 6 edges.

It has been conjectured by Pach et al. ^[13] that every k-quasi-planar topological graph with n vertices has at most c(k)n edges, where c(k) is a constant depending only on k. This conjecture is known to be true for k = 3 ^{[14] [15]} and k = 4. ^[16] It is also known to be true for convex geometric graphs (that is for geometric graphs whose vertices form the vertex set of a convex n-gon), ^[17] and for k-quasi-planar topological graphs whose edges are drawn as x-monotone curves, all of which cross a vertical line. ^{[18] [19]} The last result implies that every k-quasi-planar topological graph with n vertices, whose edges are drawn as x-monotone curves has at most c(k)n log n edges for a suitable constant c(k). For geometric graphs, this was proved earlier by Valtr. ^[20] The best known general upper bound for the number of edges of a k-quasi-planar topological graph is ${\displaystyle n\log ^{O(\log k)}n}$. ^[21] This implies that every complete topological graph with n vertices has at least ${\displaystyle n^{\frac {1}{O(\log \log n)}}}$pairwise crossing edges, which was improved to a quasi linear bound in the case of geometric graph. ^[22]

Crossing numbers

Main article: Crossing number (graph theory)

Ever since Pál Turán coined his brick factory problem ^[23] during World War II, the determination or estimation of crossing numbers of graphs has been a popular theme in graph theory and in the theory of algorithms ^[24] that is abundant with famous long standing open problems such as the Albertson conjecture, Harary-Hill's conjecture ^[25] or the still unsolved Turán's brick factory problem. ^[26] However, the publications in the subject (explicitly or implicitly) used several competing definitions of crossing numbers. This was pointed out by Pach and Tóth, ^[27] who introduced the following terminology.

Crossing number (of a graph G): The minimum number of crossing points over all drawings of G in the plane (that is, all of its representations as a topological graph) with the property that no three edges pass through the same point. It is denoted by cr(G).

Pair-crossing number: The minimum number of crossing pairs of edges over all drawings of G. It is denoted by pair-cr(G).

Odd-crossing number: The minimum number of those pairs of edges that cross an odd number of times, over all drawings of G. It is denoted by odd-cr(G).

These parameters are not unrelated. One has odd-cr(G) ≤ pair-cr(G) ≤ cr(G) for every graph G. It is known that cr(G) ≤ 2(odd-cr(G))^{2 [27]} and ${\displaystyle O(\operatorname {pcr} (G)^{\frac {3}{2}}\log ^{2}\operatorname {pcr} (G))}$ ^[28] and that there exist infinitely many graphs for which pair-cr(G) ≠ odd-cr(G). ^{[1] [29]} No examples are known for which the crossing number and the pair-crossing number are not the same. It follows from the Hanani–Tutte theorem ^{[30] [31]} that odd-cr(G) = 0 implies cr(G) = 0. It is also known that odd-cr(G) = k implies cr(G)=k for k = 1, 2, 3. ^[32] Another well researched graph parameter is the following.

Rectilinear crossing number: The minimum number of crossing points over all straight-line drawings of G in the plane (that is, all of its representations as a geometric graph) with the property that no three edges pass through the same point. It is denoted by lin-cr(G).

By definition, one has cr(G) ≤ lin-cr(G) for every graph G. It was shown by Bienstock and Dean that there are graphs with crossing number 4 and with arbitrarily large rectilinear crossing number. ^[33]

Computing the crossing number is NP-complete ^[34] in general. Therefore a large body of research focuses on its estimates. The Crossing Lemma is a cornerstone result that provides widely applicable lower bounds on the crossing number. Several interesting variants and generalizations of the Crossing Lemma are known for Jordan curves ^{[35] [36]} and degenerate crossing number ^{[37] [38]} , where the latter relates the notion of the crossing number to the graph genus.

Ramsey-type problems for geometric graphs

In traditional graph theory, a typical Ramsey-type result states that if we color the edges of a sufficiently large complete graph with a fixed number of colors, then we necessarily find a monochromatic subgraph of a certain type. ^[39] One can raise similar questions for geometric (or topological) graphs, except that now we look for monochromatic (one-colored) substructures satisfying certain geometric conditions. ^[40] One of the first results of this kind states that every complete geometric graph whose edges are colored with two colors contains a non-crossing monochromatic spanning tree . ^[41] It is also true that every such geometric graph contains ${\displaystyle \left\lceil {\frac {n+1}{3}}\right\rceil }$ disjoint edges of the same color. ^[41] The existence of a non-crossing monochromatic path of size at least cn, where c > 0 is a constant, is a long-standing open problem. It is only known that every complete geometric graph on n vertices contains a non-crossing monochromatic path of length at least ${\displaystyle n^{\frac {2}{3}}}$. ^[42]

Topological hypergraphs

If we view a topological graph as a topological realization of a 1-dimensional simplicial complex , it is natural to ask how the above extremal and Ramsey-type problems generalize to topological realizations of d-dimensional simplicial complexes. There are some initial results in this direction, but it requires further research to identify the key notions and problems. ^{[43] [44] [45]}

Two vertex disjoint simplices are said to cross if their relative interiors have a point in common. A set of k > 3 simplices strongly cross if no 2 of them share a vertex, but their relative interiors have a point common.

It is known that a set of d-dimensional simplices spanned by n points in ${\displaystyle \mathbb {R} ^{d}}$ without a pair of crossing simplices can have at most ${\displaystyle O(n^{d})}$ simplices and this bound is asymptotically tight. ^[46] This result was generalized to sets of 2-dimensional simplices in ${\displaystyle \mathbb {R} ^{2}}$ without three strongly crossing simplices. ^[47] If we forbid k strongly crossing simplices, the corresponding best known upper bound is ${\displaystyle O(n^{d+1-\delta })}$, ^[46] for some ${\displaystyle \delta =\delta (k,d)<1}$. This result follows from the colored Tverberg theorem. ^[48] It is far from the conjectured bound of ${\displaystyle O(n^{d})}$. ^[46]

For any fixed k > 1, we can select at most ${\displaystyle O(n^{\lceil {\frac {d}{2}}\rceil })}$d-dimensional simplices spanned by a set of n points in ${\displaystyle \mathbb {R} ^{d}}$ with the property that no k of them share a common interior point. ^{[46] [49]} This is asymptotically tight.

Two triangles in ${\displaystyle \mathbb {R} ^{3}}$ are said to be almost disjoint if they are disjoint or if they share only one vertex. It is an old problem of Gil Kalai and others to decide whether the largest number of almost disjoint triangles that can be chosen on some vertex set of n points in ${\displaystyle \mathbb {R} ^{3}}$ is ${\displaystyle o(n^{2})}$. It is known that there exists sets of n points for which this number is at least ${\displaystyle cn^{\frac {3}{2}}}$ for a suitable constant c > 0. ^[50]

References

1. Pelsmajer, Michael J.; Schaefer, Marcus; Štefankovič, Daniel (2008), "Odd crossing number and crossing number are not the same", Discrete and Computational Geometry , 39 (1–3): 442–454, doi: 10.1007/s00454-008-9058-x A preliminary version of these results was reviewed in Proc. of 13th International Symposium on Graph Drawing, 2005, pp. 386–396, doi: 10.1007/11618058_35
2. Hopf, Heinz; Pannwitz, Erika (1934), "Aufgabe nr. 167", Jahresbericht der Deutschen Mathematiker-Vereinigung, 43: 114
3. Lovász, László; Pach, János; Szegedy, Mario (1997), "On Conway's thrackle conjecture", Discrete and Computational Geometry , Springer, 18 (4): 369–376, doi: 10.1007/PL00009322
4. Fulek, Radoslav; Pach, János (2019), "Thrackles: An improved upper bound", Discrete Applied Mathematics , 259: 226–231, arXiv: 1708.08037 , doi:10.1016/j.dam.2018.12.025 .
5. Pach, János; Sterling, Ethan (2011), "Conway's conjecture for monotone thrackles", American Mathematical Monthly , 118 (6): 544–548, doi:10.4169/amer.math.monthly.118.06.544, MR   2812285, S2CID   17558559
6. Alon, Noga; Erdős, Paul (1989), "Disjoint edges in geometric graphs", Discrete and Computational Geometry , 4 (4): 287–290, doi: 10.1007/bf02187731
7. Černý, Jakub (2005), "Geometric graphs with no three disjoint edges", Discrete and Computational Geometry , 34 (4): 679–695, doi: 10.1007/s00454-005-1191-1
8. Pach, János; Töröcsik, Jenö (1994), "Some geometric applications of Dilworth's theorem", Discrete and Computational Geometry , 12: 1–7, doi: 10.1007/BF02574361
9. Tóth, Géza (2000), "Note on geometric graphs", Journal of Combinatorial Theory , Series A, 89 (1): 126–132, doi: 10.1006/jcta.1999.3001
10. Pach, János; Tóth, Géza (2003), "Disjoint edges in topological graphs", Combinatorial Geometry and Graph Theory: Indonesia-Japan Joint Conference, IJCCGGT 2003, Bandung, Indonesia, September 13-16, 2003, Revised Selected Papers (PDF), Lecture Notes in Computer Science, vol. 3330, Springer-Verlag, pp. 133–140, doi:10.1007/978-3-540-30540-8_15
11. Ruiz-Vargas, Andres J. (2015), "Many disjoint edges in topological graphs", in Campêlo, Manoel; Corrêa, Ricardo; Linhares-Sales, Cláudia; Sampaio, Rudini (eds.), LAGOS'15 – VIII Latin-American Algorithms, Graphs and Optimization Symposium, Electronic Notes in Discrete Mathematics, vol. 50, Elsevier, pp. 29–34, arXiv: 1412.3833 , doi:10.1016/j.endm.2015.07.006, S2CID   14865350
12. Ruiz-Vargas, Andres J. (2017), "Many disjoint edges in topological graphs", Comput. Geom. , 62: 1–13, arXiv: 1412.3833 , doi:10.1016/j.comgeo.2016.11.003
13. Pach, János; Shahrokhi, Farhad; Szegedy, Mario (1996), "Applications of the crossing number", Algorithmica , Springer, 16 (1): 111–117, doi:10.1007/BF02086610, S2CID   20375896
14. Agarwal K., Pankaj; Aronov, Boris; Pach, János; Pollack, Richard; Sharir, Micha (1997), "Quasi-planar graphs have a linear number of edges", Combinatorica , 17: 1–9, doi:10.1007/bf01196127, S2CID   8092013
15. Ackerman, Eyal; Tardos, Gábor (2007), "On the maximum number of edges in quasi-planar graphs", Journal of Combinatorial Theory , Series A, 114 (3): 563–571, doi:10.1016/j.jcta.2006.08.002
16. Ackerman, Eyal (2009), "On the maximum number of edges in topological graphs with no four pairwise crossing edges", Discrete and Computational Geometry , 41 (3): 365–375, doi: 10.1007/s00454-009-9143-9
17. Capoyleas, Vasilis; Pach, János (1992), "A Turán-type theorem on chords of a convex polygon", Journal of Combinatorial Theory , Series B, 56 (1): 9–15, doi:10.1016/0095-8956(92)90003-G
18. Suk, Andrew (2011), "k-quasi-planar graphs", Graph Drawing: 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21-23, 2011, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7034, Springer-Verlag, pp. 266–277, arXiv: 1106.0958 , doi:10.1007/978-3-642-25878-7_26, S2CID   18681576
19. Fox, Jacob; Pach, János; Suk, Andrew (2013), "The number of edges in k-quasi-planar graphs", SIAM Journal on Discrete Mathematics , 27 (1): 550–561, arXiv: 1112.2361 , doi:10.1137/110858586, S2CID   52317 .
20. Valtr, Pavel (1997), "Graph drawing with no k pairwise crossing edges", Graph Drawing: 5th International Symposium, GD '97 Rome, Italy, September 18–20, 1997, Proceedings, Lecture Notes in Computer Science, vol. 1353, Springer-Verlag, pp. 205–218
21. Fox, Jacob; Pach, János (2012), "Coloring ${\displaystyle K_{\mbox{k}}}$-free intersection graphs of geometric objects in the plane", European Journal of Combinatorics , 33 (5): 853–866, doi: 10.1016/j.ejc.2011.09.021 A preliminary version of these results was announced in Proc. of Symposium on Computational Geometry (PDF), 2008, pp. 346–354, doi:10.1145/1377676.1377735, S2CID   15138458
22. Pach, János; Rubin, Natan; Tardos, Gábor (2021), "Planar point sets determine many pairwise crossing segments", Advances in Mathematics , 386: 107779, arXiv: 1904.08845 , doi:10.1016/j.aim.2021.107779 .
23. Turán, Paul (1977), "A note of welcome", Journal of Graph Theory , 1: 7–9, doi:10.1002/jgt.3190010105
24. Garey, M. R.; Johnson, D. S. (1983), "Crossing number is NP-complete", SIAM Journal on Algebraic and Discrete Methods, 4 (3): 312–316, doi:10.1137/0604033, MR   0711340
25. Balogh, József; Lidický, Bernard; Salazar, Gelasio (2019), "Closing in on Hill's conjecture", SIAM Journal on Discrete Mathematics , 33 (3): 1261–1276, arXiv: 1711.08958 , doi:10.1137/17M1158859, S2CID   119672893
26. Schaefer, Marcus (2012), "The graph crossing number and its variants: A survey", Electronic Journal of Combinatorics , 1000: DS21–May, doi: 10.37236/2713
27. Pach, János; Tóth, Géza (2000), "Which crossing number is it anyway?", Journal of Combinatorial Theory , Series B, 80 (2): 225–246, doi: 10.1006/jctb.2000.1978
28. Matoušek, Jiří (2014), "Near-optimal separators in string graphs", Combin. Probab. Comput., vol. 23, pp. 135–139, arXiv: 1302.6482 , doi:10.1017/S0963548313000400, S2CID   2351522
29. Tóth, Géza (2008), "Note on the pair-crossing number and the odd-crossing number", Discrete and Computational Geometry , 39 (4): 791–799, doi: 10.1007/s00454-007-9024-z .
30. Chojnacki (Hanani), Chaim (Haim) (1934), "Uber wesentlich unplattbar Kurven im dreidimensionale Raume", Fundamenta Mathematicae, 23: 135–142, doi: 10.4064/fm-23-1-135-142
31. Tutte, William T. (1970), "Toward a theory of crossing numbers", Journal of Combinatorial Theory , 8: 45–53, doi:10.1016/s0021-9800(70)80007-2
32. Pelsmajer, Michael J.; Schaefer, Marcus; Štefankovič, Daniel (2007), "Removing even crossings", Journal of Combinatorial Theory , Series B, 97 (4): 489–500, doi:10.1016/j.jctb.2006.08.001
33. Bienstock, Daniel; Dean, Nathaniel (1993), "Bounds for rectilinear crossing numbers", Journal of Graph Theory , 17 (3): 333–348, doi:10.1002/jgt.3190170308
34. Garey, M. R.; Johnson, D. S. (1983). "Crossing number is NP-complete". SIAM Journal on Algebraic and Discrete Methods. 4 (3): 312–316. doi:10.1137/0604033. MR   0711340.
35. Pach, János; Rubin, Natan; Tardos, Gábor (2018), "A crossing lemma for Jordan curves", Advances in Mathematics , 331: 908–940, arXiv: 1708.02077 , doi:10.1016/j.aim.2018.03.015, S2CID   22278629
36. Pach, János; Tóth, Géza (2019), "Many touchings force many crossings", Journal of Combinatorial Theory , Series B, 137: 104–111, arXiv: 1706.06829 , doi:10.1016/j.jctb.2018.12.002
37. Ackerman, Eyal; Pinchasi, Rom (2013), "On the Degenerate Crossing Number", Discrete & Computational Geometry , 49 (3): 695–702, doi:10.1007/s00454-013-9493-1, S2CID   254030772
38. Schaefer, Marcus; Štefankovič, Daniel (2015), "The degenerate crossing number and higher-genus embeddings", Graph Drawing: 23rd International Symposium, GD 2015, Los Angeles, CA, USA, September 24-26, 2015, Revised Selected Papers, pp. 63–74, doi: 10.1007/978-3-319-27261-0_6
39. Graham, Ronald L.; Rothschild, Bruce L.; Spencer, Joel H. (1990), Ramsey Theory, Wiley
40. Károlyi, Gyula (2013), "Ramsey-type problems for geometric graphs", in Pach, J. (ed.), Thirty essays on geometric graph theory, Springer
41. Károlyi, Gyula J.; Pach, János; Tóth, Géza (1997), "Ramsey-type results for geometric graphs, I", Discrete and Computational Geometry , 18 (3): 247–255, doi: 10.1007/PL00009317
42. Károlyi, Gyula J.; Pach, János; Tóth, Géza; Valtr, Pavel (1998), "Ramsey-type results for geometric graphs, II", Discrete and Computational Geometry , 20 (3): 375–388, doi: 10.1007/PL00009391
43. Pach, János (2004), "Geometric graph theory", in Goodman, Jacob E.; O'Rourke, Joseph (eds.), Handbook of Discrete and Computational Geometry, Discrete Mathematics and Its Applications (2nd ed.), Chapman and Hall/CRC
44. Matoušek, Jiří; Tancer, Martin; Wagner, Uli (2009), "Hardness of embedding simplicial complexes in ${\displaystyle \mathbb {R} ^{d}}$", Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 855–864
45. Brass, Peter (2004), "Turán-type problems for convex geometric hypergraphs", in Pach, J. (ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, American Mathematical Society, pp. 25–33
46. Dey, Tamal K.; Pach, János (1998), "Extremal problems for geometric hypergraphs", Discrete and Computational Geometry , 19 (4): 473–484, doi: 10.1007/PL00009365
47. Suk, Andrew (2013), "A note on geometric 3-hypergraphs", in Pach, J. (ed.), Thirty Essays on Geometric Graph Theory, Springer arXiv:1010.5716v3
48. Živaljević, Rade T.; Vrećica, Siniša T. (1992), "The colored Tverberg's problem and complexes of injective functions", Journal of Combinatorial Theory , Series A, 61 (2): 309–318, doi: 10.1016/0097-3165(92)90028-S
49. Bárány, Imre; Fürédi, Zoltán (1987), "Empty simplices in Euclidean-space", Canadian Mathematical Bulletin , 30 (4): 436–445, doi:10.4153/cmb-1987-064-1, hdl: 1813/8573 , S2CID   122255929
50. Károlyi, Gyula; Solymosi, József (2002), "Almost disjoint triangles in 3-space", Discrete and Computational Geometry , 28 (4): 577–583, doi: 10.1007/s00454-002-2888-z