Abstract
We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph’s canonical polygonal schema drawn as a convex polygonal external face.
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References
Bárány, I., Rote, G.: Strictly convex drawings of planar graphs. Documenta Mathematica 11, 369–391, (2006) arXiv:cs/0507030 , http://www.math.uiuc.edu/documenta/vol-11/13.html
Becker, B., Hotz, G.: On the optimal layout of planar graphs with fixed boundary. SIAM J. Comput. 16(5), 946–972 (1987), doi:10.1137/0216061
Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Proc. 12th ACM Symp. Comput. Geom., pp. 319–328 (1996), doi:10.1145/237218.237401
Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl. 7(3), 211–223 (1997), doi:10.1142/S0218195997000144
Chrobak, M., Payne, T.H.: A linear-time algorithm for drawing a planar graph on a grid. Inf. Proc. Lett. 54(4), 241–246 (1995), doi:10.1016/0020-0190(95)00020-D
Davidson, R., Harel, D.: Drawing graphs nicely using simulated annealing. ACM Trans. Graph. 15(4), 301–331 (1996), doi:10.1145/234535.234538
Dhandapani, R.: Greedy drawings of triangulations. Discrete Comput. Geom. 43(2), 375–392 (2010), doi:10.1007/s00454-009-9235-6
di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)
Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Planar drawings of higher-genus graphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, Springer, Heidelberg (2010), doi:10.1007/978-3-642-11805-0_7
Fáry, I.: On straight-line representation of planar graphs. Acta Sci. Math. (Szeged) 11, 229–233 (1948)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990), doi:10.1007/BF02122694
Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Softw. Pract. Exp. 21(11), 1129–1164 (1991), doi:10.1002/spe.4380211102
Gajer, P., Goodrich, M.T., Kobourov, S.G.: A multi-dimensional approach to force-directed layouts of large graphs. Comput. Geom. Theory Appl. 29(1), 3–18 (2004), doi:10.1016/j.comgeo.2004.03.014
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996), doi:10.1007/BF02086606
Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an Orientable Triangulated Surface. In: Proc. 17th ACM Symp. Comput. Geom., pp. 80–89 (2001), doi:10.1145/378583.378630
Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Symp. Discrete Algorithms, pp. 138–148 (1990), http://portal.acm.org/citation.cfm?id=320191
Stein, S.K.: Convex maps. Proc. Amer. Math. Soc. 2(3), 464–466 (1951), doi:10.1090/S0002-9939-1951-0041425-5
Sugiyama, K., Misue, K.: Graph drawing by the magnetic spring model. J. Visual Lang. Comput. 6(3), 217–231 (1995), doi:10.1006/jvlc.1995.1013
Tutte, W.T.: Convex representations of graphs. Proc. London Math. Soc. 10(38), 304–320 (1960), doi:10.1112/plms/s3-10.1.304
Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13(52), 743–768 (1963), doi:10.1112/plms/s3-13.1.743
Wagner, K.: Bemerkungen zum Vierfarbenproblem. Jber. Deutsch. Math.-Verein. 46, 26–32 (1936)
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Chambers, E.W., Eppstein, D., Goodrich, M.T., Löffler, M. (2011). Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area. In: Brandes, U., Cornelsen, S. (eds) Graph Drawing. GD 2010. Lecture Notes in Computer Science, vol 6502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18469-7_12
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DOI: https://doi.org/10.1007/978-3-642-18469-7_12
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