Abstract
The
of a planar graph G is the smallest number of line segments needed for a planar straight-line drawing of G. Dujmović, Eppstein, Suderman, and Wood [CGTA’07] introduced this measure for the visual complexity of graphs. There are optimal algorithms for trees and worst-case optimal algorithms for outerplanar graphs, 2-trees, and planar 3-trees. It is known that every cubic triconnected planar n-vertex graph (except \(K_4\)) has segment number \(n/2+3\), which is the only known universal lower bound for a meaningful class of planar graphs.
We show that every triconnected planar 4-regular graph can be drawn using at most \(n+3\) segments. This bound is tight up to an additive constant, improves a previous upper bound of \(7n/4+2\) implied by a more general result of Dujmović et al., and supplements the result for cubic graphs. We also give a simple optimal algorithm for cactus graphs, generalizing the above-mentioned result for trees. We prove the first linear universal lower bounds for outerpaths, maximal outerplanar graphs, 2-trees, and planar 3-trees. This shows that the existing algorithms for these graph classes are constant-factor approximations. For maximal outerpaths, our bound is best possible and can be generalized to circular arcs.
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Notes
- 1.
A
is a connected graph where any two simple cycles share at most one vertex.
is a connected graph where any two simple cycles share at most one vertex.References
Adnan, M.A.: Minimum segment drawings of outerplanar graphs. Master’s thesis, Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka (2008). http://lib.buet.ac.bd:8080/xmlui/bitstream/handle/123456789/1565/Full%20%20Thesis%20.pdf?sequence=1 &isAllowed=y
Broersma, H.J., Duijvestijn, A.J.W., Göbel, F.: Generating all 3-connected 4-regular planar graphs from the octahedron graph. J. Graph Theory 17(5), 613–620 (1993). https://doi.org/10.1002/jgt.3190170508
Chaplick, Steven, Fleszar, Krzysztof, Lipp, Fabian, Ravsky, Alexander, Verbitsky, Oleg, Wolff, Alexander: The complexity of drawing graphs on few lines and few planes. In: Ellen, F., Kolokolova, A., Sack, J.-R. (eds.) WADS 2017. LNCS, vol. 10389, pp. 265–276. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62127-2_23
Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: Drawinggraphs on few lines and few planes. J. Comput. Geom 11(1), 433–475 (2020). https://doi.org/10.20382/jocg.v11i1a17
Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. Theory Appl. 38(3), 194–212 (2007). https://doi.org/10.1016/j.comgeo.2006.09.002
Durocher, S., Mondal, D.: Drawing plane triangulations with few segments. Comput. Geom. Theory Appl. 77, 27–39 (2019). https://doi.org/10.1016/j.comgeo.2018.02.003
Goeßmann, I., et al.: The segment number: Algorithms and universal lower bounds for some classes of planar graphs. arXiv preprint (2022). https://arxiv.org/abs/2202.11604
Hong, S., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discret. Appl. Math. 156(12), 2368–2380 (2008). https://doi.org/10.1016/j.dam.2007.10.012
Hong, S., Nagamochi, H.: Convex drawings of hierarchical planar graphs and clustered planar graphs. J. Discrete Algorithms 8(3), 282–295 (2010). https://doi.org/10.1016/j.jda.2009.05.003
Hültenschmidt, G., Kindermann, P., Meulemans, W., Schulz, A.: Drawing planar graphs with few geometric primitives. J. Graph Alg. Appl. 22(2), 357–387 (2018). https://doi.org/10.7155/jgaa.00473
Igamberdiev, A., Meulemans, W., Schulz, A.: Drawing planar cubic 3-connected graphs with few segments: Algorithms & experiments. J. Graph Algorithms Appl. 21(4), 561–588 (2017). https://doi.org/10.7155/jgaa.00430
Kindermann, P., Mchedlidze, T., Schneck, T., Symvonis, A.: Drawing planar graphs with few segments on a polynomial grid. In: Archambault, D., Tóth, C.D. (eds.) GD 2019. LNCS, vol. 11904, pp. 416–429. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-35802-0_32
Kindermann, P., Meulemans, W., Schulz, A.: Experimental analysis of the accessibility of drawings with few segments. J. Graph Alg. Appl. 22(3), 501–518 (2018). https://doi.org/10.7155/jgaa.00474
Kleist, L., Klemz, B., Lubiw, A., Schlipf, L., Staals, F., Strash, D.: Convexity-increasing morphs of planar graphs. Comput. Geom. 84, 69–88 (2019). https://doi.org/10.1016/j.comgeo.2019.07.007
Klemz, B.: Convex drawings of hierarchical graphs in linear time, with applications to planar graph morphing. In: Mutzel, P., Pagh, R., Herman, G., (eds.) Proceedings of 29th Annual European Symposium on Algorithms (ESA 2021), vol. 204 of LIPIcs, pp. 57:1–57:15. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021). https://doi.org/10.4230/LIPIcs.ESA.2021.57
Kryven, M., Ravsky, A., Wolff, A.: Drawing graphs on few circles and few spheres. J. Graph Alg. Appl. 23(2), 371–391 (2019). https://doi.org/10.7155/jgaa.00495
Mondal, D., Nishat, R.I., Biswas, S., Rahman, M.: Minimum-segment convex drawings of 3-connected cubic plane graphs. J. Comb. Optim. 25(3), 460–480 (2013). https://doi.org/10.1007/s10878-011-9390-6
Okamoto, Y., Ravsky, A., Wolff, A.: Variants of the segment number of a graph. In: Archambault, D., Tóth, C.D. (eds.) GD 2019. LNCS, vol. 11904, pp. 430–443. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-35802-0_33
Samee, M.A.H., Alam, M.J., Adnan, M.A., Rahman, M.S.: Minimum segment drawings of series-parallel graphs with the maximum degree three. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 408–419. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00219-9_40
Schulz, A.: Drawing graphs with few arcs. J. Graph Alg. Appl. 19(1), 393–412 (2015). https://doi.org/10.7155/jgaa.00366
Thomassen, C.: Plane representations of graphs. In: Bondy, J.A., Murty, U.S.R., (eds.) Progress in Graph Theory, pp. 43–69. Academic Press (1984)
Tutte, W.T.: Convex representations of graphs. Proc. London Math. Soc. s3–10(1), 304–320 (1960). https://doi.org/10.1112/plms/s3-10.1.304
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Goeßmann, I. et al. (2022). The Segment Number: Algorithms and Universal Lower Bounds for Some Classes of Planar Graphs. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_20
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