Abstract
Symmetry is one of the most important aesthetic criteria in Graph Drawing which can reveal the hidden structure in the graph. Convex drawing is a straight-line drawing where every facial cycle is drawn as a convex polygon.
In this paper, we prove that given an internally triconnected plane graph with symmetries, there exists a convex drawing of the graph which displays the given symmetries. We present a linear-time algorithm for constructing symmetric convex drawings of internally triconnected planar graphs.
This is an extension of a classical result due to Tutte (Proc. Lond. Math. Soc. 10(3):304–320, 1960; Proc. Lond. Math. Soc. 13:743–768, 1963) who proved that every triconnected plane graph with a given convex polygon as a boundary admits a convex drawing. Note that Tutte’s barycenter mapping method can be implemented in O(n 1.5) time and O(nlog n) space at best (Lipton et al. in SIAM J. Numer. Anal. 16:346–358, 1979).
Our divide and conquer algorithm explicitly exploits the fundamental properties of symmetric drawing, which consists of congruent drawings of isomorphic subgraphs. We first find an isomorphic subgraph of a given symmetric plane graph, and compute an angle-constrained convex drawing of the subgraph. Finally, a symmetric convex drawing of the given graph is constructed by merging repetitive copies of the congruent drawings of isomorphic subgraphs. For this purpose, we define a new problem of angle-constrained convex drawing of plane graphs, where some of outer vertices have angle constraints.
Our results also imply that there is a linear-time algorithm that constructs maximally symmetric convex drawings of triconnected planar graphs. Previous algorithm (Hong et al. in Discrete Comput. Geom. 36:283–311, 2006) constructs symmetric drawings of triconnected planar graphs with straight-lines.
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References
Abelson, D., Hong, S., Taylor, D.E.: A Group-theoretic method for drawing graphs symmetrically. In: Graph Drawing (Proc. of GD 2002). Lecture Notes in Computer Science, vol. 2265, pp. 86–97. Springer, Berlin (2003)
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North Holland, Amsterdam (1976)
Bonichon, N., Felsner, S., Mosbah, M.: Convex drawings of 3-connected plane graphs. In: Graph Drawing (Proc. of GD 2004). Lecture Notes in Computer Science, vol. 3383, pp. 60–70. Springer, Berlin (2005)
Buchheim, C., Junger, M.: Detecting symmetries by branch and cut. In: Graph Drawing (Proc. of GD 2001). Lecture Notes in Computer Science, pp. 178–188. Springer, Berlin (2001)
Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear algorithms for convex drawings of planar graphs. In: Progress in Graph Theory, pp. 153–173. Academic Press, San Diego (1984)
Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Int. J. Comput. Geom. Appl. 7, 211–223 (1997)
Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Proc. of the 12th Annual Symposium on Computational Geometry (SoCG 1996), pp. 319–328. ACM, New York (1996)
Cook, S.A., Reckhow, R.A.: Time bounded random access machines. J. Comput. Syst. Sci. 7, 354–375 (1976)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, New York (1999)
de Fraysseix, H.: An heuristic for graph symmetry detection. In: Graph Drawing (Proc. of GD 1999). Lecture Notes in Computer Science, vol. 1731, pp. 276–285. Springer, Berlin (2000)
Eades, P., Lin, X.: Spring algorithms and symmetry. Theor. Comput. Sci. 240(2), 379–405 (2000)
Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)
Hong, S., Eades, P.: Symmetric layout of disconnected graphs. In: Algorithms and Computation (Proc. of ISAAC 2003). Lecture Notes in Computer Science, vol. 2906, pp. 405–414. Springer, Berlin (2003)
Hong, S., Eades, P.: Drawing planar graphs symmetrically II: biconnected planar graphs. Algorithmica 42(2), 159–197 (2005)
Hong, S., Eades, P.: Drawing planar graphs symmetrically III: oneconnected planar graphs. Algorithmica 44(1), 67–100 (2006)
Hong, S., Nagamochi, H.: Convex drawings of hierarchical plane graphs. In: Proc. of the 17th Australasian Workshop on Combinatorial Algorithms (AWOCA 2006), Uluru, NT, 13–16 July 2006
Hong, S., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discrete Appl. Math. 156, 2368–2380 (2008)
Hong, S., Eades, P., Lee, S.: Drawing series parallel digraphs symmetrically. Comput. Geom. Theory Appl. 17(3–4), 165–188 (2000)
Hong, S., McKay, B., Eades, P.: A linear time algorithm for constructing maximally symmetric straight line drawings of triconnected planar graphs. Discrete Comput. Geom. 36, 283–311 (2006)
Kamada, A., Miura, K., Nishizeki, T.: Convex grid drawings of plane graphs with rectangular contours. In: Proc. of ISAAC 2006. Lecture Notes in Computer Science, pp. 131–140. Springer, Berlin (2006)
Lin, X.: Analysis of algorithms for drawing graphs. Ph.D. Thesis, University of Queensland (1992)
Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM J. Numer. Anal. 16, 346–358 (1979)
Lipton, R.J., North, S.C., Sandberg, J.S.: A method for drawing graphs. In: Proc. of Annual Symposium on Computational Geometry (SoCG), pp. 153–160. ACM, New York (1985)
Liskovets, V.A.: A reductive technique for enumerating non-isomorphic planar maps. Discrete Math. 156, 197–217 (1996)
Lubiw, A.: Some NP-complete problems similar to graph isomorphism. SIAM J. Comput. 10(1), 11–21 (1981)
Mani, P.: Automorphismen von Polyedrischen Graphen. Math. Ann. 192, 279–303 (1971)
Manning, J.: Geometric symmetry in graphs. Ph.D. Thesis, Purdue Univ. (1990)
Manning, J., Atallah, M.J.: Fast detection and display of symmetry in trees. Congressus Numerantium 64, 159–169 (1988)
Manning, J., Atallah, M.J.: Fast detection and display of symmetry in quterplanar graphs. Discrete Appl. Math. 39, 13–35 (1992)
Miura, K., Azuma, M., Nishizeki, T.: Convex drawings of plane graphs of minimum outer apices. Int. J. Found. Comput. Sci. 17, 1115–1128 (2006)
Miura, K., Nakano, S., Nishizeki, T.: Convex grid drawings of four-connected plane graphs. Int. J. Found. Comput. Sci. 17(5), 1031–1060 (2006)
Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: Graph Drawing (Proc. of GD 1997). Lecture Notes in Computer Science, vol. 1353, pp. 248–259. Springer, Berlin (1998)
Rote, G.: Strictly convex drawings of planar graphs. In: Proc. of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pp. 728–734. SIAM, Philadelphia (2005)
Tarjan, R.E.: Data Structures and Network Algorithms. SIAM, Philadelphia (1983)
Thomassen, C.: Plane representations of graphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Progress in Graph Theory, pp. 43–69. Academic Press, San Diego (1984)
Tutte, W.T.: Convex representations of graphs. Proc. Lond. Math. Soc. 10(3), 304–320 (1960)
Tutte, W.T.: How to draw a graph. Proc. Lond. Math. Soc. 13, 743–768 (1963)
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Hong, SH., Nagamochi, H. A Linear-Time Algorithm for Symmetric Convex Drawings of Internally Triconnected Plane Graphs. Algorithmica 58, 433–460 (2010). https://doi.org/10.1007/s00453-008-9275-y
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DOI: https://doi.org/10.1007/s00453-008-9275-y