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RAC drawing

From Wikipedia, the free encyclopedia
RAC drawings of the complete graph K5 and the complete bipartite graph K3,4

In graph drawing, a RAC drawing of a graph is a drawing in which the vertices are represented as points, the edges are represented as straight line segments or polylines, at most two edges cross at any point, and when two edges cross they do so at right angles to each other. In the name of this drawing style, "RAC" stands for "right angle crossing".

The right-angle crossing style and the name "RAC drawing" for this style were both formulated by Didimo, Eades & Liotta (2009),[1] motivated by previous user studies showing that crossings with large angles are much less harmful to the readability of drawings than shallow crossings.[2] Even for planar graphs, allowing some right-angle crossings in a drawing of the graph can significantly improve measures of the drawing quality such as its area or angular resolution.[3]

Examples

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The complete graph K5 has a RAC drawing with straight edges, but K6 does not. Every 6-vertex RAC drawing has at most 14 edges, but K6 has 15 edges, too many to have a RAC drawing.[1]

A complete bipartite graph Ka,b has a RAC drawing with straight edges if and only if either min(a,b) ≤ 2 or a + b ≤ 7. If min(a,b) ≤ 2, then the graph is a planar graph, and (by Fáry's theorem) every planar graph has a straight-line drawing with no crossings. Such a drawing is automatically a RAC drawing. The only two cases remaining are the graphs K3,3 and K3,4. A drawing of K3,4 is shown; K3,3 can be formed from it by deleting one vertex. Neither of the next two larger graphs, K4,4 and K3,5, has a RAC drawing.[4]

Edges and bends

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If an n-vertex graph (n ≥ 4) has a RAC drawing with straight edges, it can have at most 4n − 10 edges. This is tight: there exist RAC-drawable graphs with exactly 4n − 10 edges.[1] For drawings with polyline edges, the bound on the number of edges in the graph depends on the number of bends that are allowed per edge. The graphs that have RAC drawings with one or two bends per edge have O(n) edges; more specifically, with one bend there are at most 5.5n edges[5] and with two bends there are at most 74.2n edges.[6] Every graph has a RAC drawing with three bends per edge.[1]

Relation to 1-planarity

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A graph is 1-planar if it has a drawing with at most one crossing per edge. Intuitively, this restriction makes it easier to cause this crossing to be at right angles, and the 4n − 10 bound on the number of edges of straight-line RAC drawings is close to the bounds of 4n − 8 on the number of edges in a 1-planar graph, and of 4n − 9 on the number of edges in a straight-line 1-planar graph. Every RAC drawing with 4n − 10 edges is 1-planar.[7][8] Additionally, every outer-1-planar graph (that is, a graph drawn with one crossing per edge with all vertices on the outer face of the drawing) has a RAC drawing.[9] However, there exist 1-planar graphs with 4n − 10 edges that do not have RAC drawings.[7]

Computational complexity

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It is NP-hard to determine whether a given graph has a RAC drawing with straight edges,[10] even if the input graph is 1-planar and the output RAC drawing must be 1-planar as well.[11] More specifically, RAC drawing is complete for the existential theory of the reals.[12] The RAC drawing problem remains NP-hard for upward drawing of directed acyclic graphs.[13] However, in the special case of outer-1-planar graphs, a RAC drawing can be constructed in linear time.[14]

References

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  1. ^ a b c d Didimo, Walter; Eades, Peter; Liotta, Giuseppe (2009), "Drawing graphs with right angle crossings", Algorithms and Data Structures: 11th International Symposium, WADS 2009, Banff, Canada, August 21–23, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5664, pp. 206–217, doi:10.1007/978-3-642-03367-4_19, ISBN 978-3-642-03366-7.
  2. ^ Huang, Weidong; Hong, Seok-Hee; Eades, Peter (2008), "Effects of crossing angles", IEEE Pacific Visualization Symposium (PacificVIS '08), pp. 41–46, doi:10.1109/PACIFICVIS.2008.4475457, ISBN 978-1-4244-1966-1.
  3. ^ van Kreveld, Marc (2011), "The quality ratio of RAC drawings and planar drawings of planar graphs", Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21–24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6502, pp. 371–376, doi:10.1007/978-3-642-18469-7_34, ISBN 978-3-642-18468-0.
  4. ^ Didimo, Walter; Eades, Peter; Liotta, Giuseppe (2010), "A characterization of complete bipartite RAC graphs", Information Processing Letters, 110 (16): 687–691, doi:10.1016/j.ipl.2010.05.023, MR 2676805.
  5. ^ Angelini, Patrizio; Bekos, Michael; Förster, Henry; Kaufmann, Michael (2018), On RAC Drawings of Graphs with one Bend per Edge, arXiv:1808.10470
  6. ^ Arikushi, Karin; Fulek, Radoslav; Keszegh, Balázs; Morić, Filip; Tóth, Csaba D. (2012), "Graphs that admit right angle crossing drawings", Computational Geometry Theory & Applications, 45 (4): 169–177, arXiv:1001.3117, doi:10.1016/j.comgeo.2011.11.008, MR 2876688.
  7. ^ a b Eades, Peter; Liotta, Giuseppe (2013), "Right angle crossing graphs and 1-planarity", Discrete Applied Mathematics, 161 (7–8): 961–969, doi:10.1016/j.dam.2012.11.019, MR 3030582.
  8. ^ Ackerman, Eyal (2014), "A note on 1-planar graphs", Discrete Applied Mathematics, 175: 104–108, doi:10.1016/j.dam.2014.05.025, MR 3223912.
  9. ^ Dehkordi, Hooman Reisi; Eades, Peter (2012), "Every outer-1-plane graph has a right angle crossing drawing", International Journal of Computational Geometry & Applications, 22 (6): 543–557, doi:10.1142/S021819591250015X, MR 3042921.
  10. ^ Argyriou, Evmorfia N.; Bekos, Michael A.; Symvonis, Antonios (2011), "The straight-line RAC drawing problem is NP-hard", SOFSEM 2011: 37th Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 22-28, 2011, Proceedings, Lecture Notes in Computer Science, vol. 6543, pp. 74–85, arXiv:1009.5227, Bibcode:2011LNCS.6543...74A, doi:10.1007/978-3-642-18381-2_6, ISBN 978-3-642-18380-5
  11. ^ Bekos, Michael A.; Didimo, Walter; Liotta, Giuseppe; Mehrabi, Saeed; Montecchiani, Fabrizio (2017), "On RAC drawings of 1-planar graphs", Theoretical Computer Science, 689: 48–57, arXiv:1608.08418, doi:10.1016/j.tcs.2017.05.039
  12. ^ Schaefer, Marcus (2021), "RAC-drawability is -complete", Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021), arXiv:2107.11663
  13. ^ Angelini, Patrizio; Cittadini, Luca; Di Battista, Giuseppe; Didimo, Walter; Frati, Fabrizio; Kaufmann, Michael; Symvonis, Antonios (2010), "On the perspectives opened by right angle crossing drawings", Graph Drawing: 17th International Symposium, GD 2009, Chicago, IL, USA, September 22–25, 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5849, pp. 21–32, doi:10.1007/978-3-642-11805-0_5, ISBN 978-3-642-11804-3.
  14. ^ Auer, Christopher; Bachmaier, Christian; Brandenburg, Franz J.; Hanauer, Kathrin; Gleißner, Andreas; Neuwirth, Daniel; Reislhuber, Josef (2013), "Recognizing Outer 1-Planar Graphs in Linear Time", Graph Drawing LNCS, Lecture Notes in Computer Science, 8284: 107–118, doi:10.1007/978-3-319-03841-4, ISBN 978-3-319-03840-7
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