Slope number

In graph drawing and geometric graph theory, the slope number of a graph is the minimum possible number of distinct slopes of edges in a drawing of the graph in which vertices are represented as points in the Euclidean plane and edges are represented as line segments that do not pass through any non-incident vertex.

A drawing of the Petersen graph with slope number 3

Complete graphs

Although closely related problems in discrete geometry had been studied earlier, e.g. by Scott (1970) and Jamison (1984), the problem of determining the slope number of a graph was introduced by Wade & Chu (1994), who showed that the slope number of an n-vertex complete graph K_n is exactly n. A drawing with this slope number may be formed by placing the vertices of the graph on a regular polygon.

Relation to degree

The slope number of a graph of maximum degree d is clearly at least ${\displaystyle \lceil d/2\rceil }$, because at most two of the incident edges at a degree-d vertex can share a slope. More precisely, the slope number is at least equal to the linear arboricity of the graph, since the edges of a single slope must form a linear forest, and the linear arboricity in turn is at least ${\displaystyle \lceil d/2\rceil }$.

Unsolved problem in mathematics: Do the graphs of maximum degree four have bounded slope number? (more unsolved problems in mathematics)

There exist graphs with maximum degree five that have arbitrarily large slope number.[1] However, every graph of maximum degree three has slope number at most four;[2] the result of Wade & Chu (1994) for the complete graph K₄ shows that this is tight. Not every set of four slopes is suitable for drawing all degree-3 graphs: a set of slopes is suitable for this purpose if and only it forms the slopes of the sides and diagonals of a parallelogram. In particular, any degree 3 graph can be drawn so that its edges are either axis-parallel or parallel to the main diagonals of the integer lattice.[3] It is not known whether graphs of maximum degree four have bounded or unbounded slope number.[4]

The method of Keszegh, Pach & Pálvölgyi (2011) for combining circle packings and quadtrees to achieve bounded slope number for planar graphs with bounded degree

Planar graphs

As Keszegh, Pach & Pálvölgyi (2011) showed, every planar graph has a planar straight-line drawing in which the number of distinct slopes is a function of the degree of the graph. Their proof follows a construction of Malitz & Papakostas (1994) for bounding the angular resolution of planar graphs as a function of degree, by completing the graph to a maximal planar graph without increasing its degree by more than a constant factor, and applying the circle packing theorem to represent this augmented graph as a collection of tangent circles. If the degree of the initial graph is bounded, the ratio between the radii of adjacent circles in the packing will also be bounded by the ring lemma,[5] which in turn implies that using a quadtree to place each graph vertex on a point within its circle will produce slopes that are ratios of small integers. The number of distinct slopes produced by this construction is exponential in the degree of the graph.

Complexity

It is NP-complete to determine whether a graph has slope number two.[6] From this, it follows that it is NP-hard to determine the slope number of an arbitrary graph, or to approximate it with an approximation ratio better than 3/2.

It is also NP-complete to determine whether a planar graph has a planar drawing with slope number two,[7] and hard for the existential theory of the reals to determine the minimum slope number of a planar drawing.[8]

Notes

1. Proved independently by Barát, Matoušek & Wood (2006) and Pach & Pálvölgyi (2006), solving a problem posed by Dujmović, Suderman & Wood (2005). See theorems 5.1 and 5.2 of Pach & Sharir (2009).
2. Mukkamala & Szegedy (2009), improving an earlier result of Keszegh et al. (2008); theorem 5.3 of Pach & Sharir (2009).
3. Mukkamala & Pálvölgyi (2012).
4. Pach & Sharir (2009).
5. Hansen (1988).
6. Formann et al. (1993); Eades, Hong & Poon (2010); Maňuch et al. (2011).
7. Garg & Tamassia (2001).
8. Hoffmann (2016).

References

• Barát, János; Matoušek, Jiří; Wood, David R. (2006), "Bounded-degree graphs have arbitrarily large geometric thickness", Electronic Journal of Combinatorics, 13 (1): R3, MR 2200531.
• Dujmović, Vida; Suderman, Matthew; Wood, David R. (2005), "Really straight graph drawings", in Pach, János (ed.), Graph Drawing: 12th International Symposium, GD 2004, New York, NY, USA, September 29-October 2, 2004, Revised Selected Papers, Lecture Notes in Computer Science, 3383, Berlin: Springer-Verlag, pp. 122–132, arXiv:cs/0405112, doi:10.1007/978-3-540-31843-9_14.
• Eades, Peter; Hong, Seok-Hee; Poon, Sheung-Hung (2010), "On rectilinear drawing of graphs", in Eppstein, David; Gansner, Emden R. (eds.), Graph Drawing: 17th International Symposium, GD 2009, Chicago, IL, USA, September 22-25, 2009, Revised Papers, Lecture Notes in Computer Science, 5849, Berlin: Springer, pp. 232–243, doi:10.1007/978-3-642-11805-0_23, MR 2680455.
• Formann, M.; Hagerup, T.; Haralambides, J.; Kaufmann, M.; Leighton, F. T.; Symvonis, A.; Welzl, E.; Woeginger, G. (1993), "Drawing graphs in the plane with high resolution", SIAM Journal on Computing, 22 (5): 1035–1052, doi:10.1137/0222063, MR 1237161.
• Garg, Ashim; Tamassia, Roberto (2001), "On the computational complexity of upward and rectilinear planarity testing", SIAM Journal on Computing, 31 (2): 601–625, doi:10.1137/S0097539794277123, MR 1861292.
• Hansen, Lowell J. (1988), "On the Rodin and Sullivan ring lemma", Complex Variables, Theory and Application, 10 (1): 23–30, doi:10.1080/17476938808814284, MR 0946096.
• Hoffmann, Udo (2016), "The planar slope number", Proceedings of the 28th Canadian Conference on Computational Geometry (CCCG 2016).
• Jamison, Robert E. (1984), "Planar configurations which determine few slopes", Geometriae Dedicata, 16 (1): 17–34, doi:10.1007/BF00147419, MR 0757792.
• Keszegh, Balázs; Pach, János; Pálvölgyi, Dömötör (2011), "Drawing planar graphs of bounded degree with few slopes", in Brandes, Ulrik; Cornelsen, Sabine (eds.), Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, 6502, Heidelberg: Springer, pp. 293–304, arXiv:1009.1315, doi:10.1007/978-3-642-18469-7_27, MR 2781274.
• Keszegh, Balázs; Pach, János; Pálvölgyi, Dömötör; Tóth, Géza (2008), "Drawing cubic graphs with at most five slopes", Computational Geometry: Theory and Applications, 40 (2): 138–147, doi:10.1016/j.comgeo.2007.05.003, MR 2400539.
• Malitz, Seth; Papakostas, Achilleas (1994), "On the angular resolution of planar graphs", SIAM Journal on Discrete Mathematics, 7 (2): 172–183, doi:10.1137/S0895480193242931, MR 1271989.
• Maňuch, Ján; Patterson, Murray; Poon, Sheung-Hung; Thachuk, Chris (2011), "Complexity of finding non-planar rectilinear drawings of graphs", in Brandes, Ulrik; Cornelsen, Sabine (eds.), Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, 6502, Heidelberg: Springer, pp. 305–316, doi:10.1007/978-3-642-18469-7_28, hdl:10281/217381, MR 2781275.
• Mukkamala, Padmini; Szegedy, Mario (2009), "Geometric representation of cubic graphs with four directions", Computational Geometry: Theory and Applications, 42 (9): 842–851, doi:10.1016/j.comgeo.2009.01.005, MR 2543806.
• Mukkamala, Padmini; Pálvölgyi, Dömötör (2012), "Drawing cubic graphs with the four basic slopes", in van Kreveld, Marc; Speckmann, Bettina (eds.), Graph Drawing: 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21-23, 2011, Revised Selected Papers, Lecture Notes in Computer Science, 7034, Springer, pp. 254–265, arXiv:1106.1973, doi:10.1007/978-3-642-25878-7_25.
• Pach, János; Pálvölgyi, Dömötör (2006), "Bounded-degree graphs can have arbitrarily large slope numbers", Electronic Journal of Combinatorics, 13 (1): N1, MR 2200545.
• Pach, János; Sharir, Micha (2009), "5.5 Angular resolution and slopes", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, 152, American Mathematical Society, pp. 126–127.
• Scott, P. R. (1970), "On the sets of directions determined by n points", American Mathematical Monthly, 77: 502–505, doi:10.2307/2317384, MR 0262933.
• Wade, G. A.; Chu, J.-H. (1994), "Drawability of complete graphs using a minimal slope set", The Computer Journal, 37 (2): 139–142, doi:10.1093/comjnl/37.2.139.