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2013 | OriginalPaper | Buchkapitel

Drawing Trees, Outerplanar Graphs, Series-Parallel Graphs, and Planar Graphs in a Small Area

verfasst von : Giuseppe Di Battista, Fabrizio Frati

Erschienen in: Thirty Essays on Geometric Graph Theory

Verlag: Springer New York

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Abstract

In this chapter, we survey algorithms and bounds for constructing planar drawings of graphs in a small area.

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Literatur
1.
Zurück zum Zitat G.E. Andrews, A lower bound for the volumes of strictly convex bodies with many boundary points. Trans. Am. Math. Soc. 106, 270–279 (1963)MATHCrossRef G.E. Andrews, A lower bound for the volumes of strictly convex bodies with many boundary points. Trans. Am. Math. Soc. 106, 270–279 (1963)MATHCrossRef
2.
Zurück zum Zitat P. Angelini, G. Di Battista, F. Frati, Succinct greedy drawings do not always exist, in Graph Drawing (GD’09), ed. by D. Eppstein, E.R. Gansner. LNCS, Springer, vol. 5849 (2010) pp. 171–182 P. Angelini, G. Di Battista, F. Frati, Succinct greedy drawings do not always exist, in Graph Drawing (GD’09), ed. by D. Eppstein, E.R. Gansner. LNCS, Springer, vol. 5849 (2010) pp. 171–182
3.
Zurück zum Zitat P. Angelini, T. Bruckdorfer, M. Chiesa, F. Frati, M. Kaufmann, C. Squarcella, On the area requirements of Euclidean minimum spanning trees (2011), manuscript P. Angelini, T. Bruckdorfer, M. Chiesa, F. Frati, M. Kaufmann, C. Squarcella, On the area requirements of Euclidean minimum spanning trees (2011), manuscript
4.
Zurück zum Zitat P. Angelini, F. Frati, L. Grilli, An algorithm to construct greedy drawings of triangulations. J. Graph Algorithm. Appl. 14(1), 19–51 (2010)MathSciNetMATHCrossRef P. Angelini, F. Frati, L. Grilli, An algorithm to construct greedy drawings of triangulations. J. Graph Algorithm. Appl. 14(1), 19–51 (2010)MathSciNetMATHCrossRef
5.
Zurück zum Zitat P. Angelini, F. Frati, M. Kaufmann, Straight-line rectangular drawings of clustered graphs. Discr. Comput. Geom. 45(1), 88–140 (2011)MathSciNetMATHCrossRef P. Angelini, F. Frati, M. Kaufmann, Straight-line rectangular drawings of clustered graphs. Discr. Comput. Geom. 45(1), 88–140 (2011)MathSciNetMATHCrossRef
6.
Zurück zum Zitat P. Angelini, F. Frati, M. Patrignani, Splitting clusters to get c-planarity, in Graph Drawing (GD’09), ed. by D. Eppstein, E.R. Gansner. LNCS, Springer, vol. 5849 (2010), pp. 57–68 P. Angelini, F. Frati, M. Patrignani, Splitting clusters to get c-planarity, in Graph Drawing (GD’09), ed. by D. Eppstein, E.R. Gansner. LNCS, Springer, vol. 5849 (2010), pp. 57–68
7.
Zurück zum Zitat I. Bárány, J. Pach, On the number of convex lattice polygons. Comb. Prob. Comput. 1, 295–302 (1992)MATHCrossRef I. Bárány, J. Pach, On the number of convex lattice polygons. Comb. Prob. Comput. 1, 295–302 (1992)MATHCrossRef
8.
Zurück zum Zitat I. Bárány, G. Rote, Strictly convex drawings of planar graphs. Doc. Math. 11, 369–391 (2006)MathSciNetMATH I. Bárány, G. Rote, Strictly convex drawings of planar graphs. Doc. Math. 11, 369–391 (2006)MathSciNetMATH
10.
Zurück zum Zitat P. Bertolazzi, R.F. Cohen, G. Di Battista, R. Tamassia, I.G. Tollis, How to draw a series-parallel digraph. Int. J. Comput. Geom. Appl. 4(4), 385–402 (1994)MATHCrossRef P. Bertolazzi, R.F. Cohen, G. Di Battista, R. Tamassia, I.G. Tollis, How to draw a series-parallel digraph. Int. J. Comput. Geom. Appl. 4(4), 385–402 (1994)MATHCrossRef
11.
Zurück zum Zitat P. Bertolazzi, G. Di Battista, G. Liotta, C. Mannino, Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)MathSciNetMATHCrossRef P. Bertolazzi, G. Di Battista, G. Liotta, C. Mannino, Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)MathSciNetMATHCrossRef
12.
Zurück zum Zitat T.C. Biedl, Drawing outer-planar graphs in O(nlogn) area, in Graph Drawing (GD’02), ed. by M.T. Goodrich. LNCS, Springer, vol. 2528 (2002), pp. 54–65 T.C. Biedl, Drawing outer-planar graphs in O(nlogn) area, in Graph Drawing (GD’02), ed. by M.T. Goodrich. LNCS, Springer, vol. 2528 (2002), pp. 54–65
13.
Zurück zum Zitat T.C. Biedl, Small drawings of outerplanar graphs, series-parallel graphs, and other planar graphs. Discr. Comput. Geom. 45(1), 141–160 (2011)MathSciNetMATHCrossRef T.C. Biedl, Small drawings of outerplanar graphs, series-parallel graphs, and other planar graphs. Discr. Comput. Geom. 45(1), 141–160 (2011)MathSciNetMATHCrossRef
14.
Zurück zum Zitat T.C. Biedl, F.J. Brandenburg, Drawing planar bipartite graphs with small area, in Canadian Conference on Computational Geometry (CCCG’05), University of Windsor, Ontario, Canada, 2005, pp. 105–108 T.C. Biedl, F.J. Brandenburg, Drawing planar bipartite graphs with small area, in Canadian Conference on Computational Geometry (CCCG’05), University of Windsor, Ontario, Canada, 2005, pp. 105–108
15.
16.
Zurück zum Zitat T.C. Biedl, G. Kant, M. Kaufmann, On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)MathSciNetMATHCrossRef T.C. Biedl, G. Kant, M. Kaufmann, On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)MathSciNetMATHCrossRef
17.
18.
Zurück zum Zitat N. Bonichon, B. Le Saëc, M. Mosbah, Wagner’s theorem on realizers, in Automata, Languages and Programming (ICALP’02), ed. by P. Widmayer, F. Triguero Ruiz, R.M. Bueno, M. Hennessy, S. Eidenbenz, R. Conejo. LNCS, Springer, vol. 2380 (2002), pp. 1043–1053 N. Bonichon, B. Le Saëc, M. Mosbah, Wagner’s theorem on realizers, in Automata, Languages and Programming (ICALP’02), ed. by P. Widmayer, F. Triguero Ruiz, R.M. Bueno, M. Hennessy, S. Eidenbenz, R. Conejo. LNCS, Springer, vol. 2380 (2002), pp. 1043–1053
20.
Zurück zum Zitat F.J. Brandenburg, Drawing planar graphs on \(\frac{8} {9}{n}^{2}\) area. Electron. Notes Discr. Math. 31, 37–40 (2008)MathSciNetCrossRef F.J. Brandenburg, Drawing planar graphs on \(\frac{8} {9}{n}^{2}\) area. Electron. Notes Discr. Math. 31, 37–40 (2008)MathSciNetCrossRef
21.
Zurück zum Zitat A. Brocot, Calcul des rouages par approximation, nouvelle methode. Rev. Chronom. 6, 186–194 (1860) A. Brocot, Calcul des rouages par approximation, nouvelle methode. Rev. Chronom. 6, 186–194 (1860)
22.
Zurück zum Zitat L. Cao, A. Strelzoff, J.Z. Sun, On succinctness of geometric greedy routing in Euclidean plane, in Pervasive Systems, Algorithms, and Networks (ISPAN’09), Kaohsiung, Taiwan, 2009, pp. 326–331 L. Cao, A. Strelzoff, J.Z. Sun, On succinctness of geometric greedy routing in Euclidean plane, in Pervasive Systems, Algorithms, and Networks (ISPAN’09), Kaohsiung, Taiwan, 2009, pp. 326–331
23.
Zurück zum Zitat T.M. Chan, M.T. Goodrich, S. Rao Kosaraju, R. Tamassia, Optimizing area and aspect ratio in straight-line orthogonal tree drawings. Comput. Geom.: Theor. Appl. 23(2), 153–162 (2002) T.M. Chan, M.T. Goodrich, S. Rao Kosaraju, R. Tamassia, Optimizing area and aspect ratio in straight-line orthogonal tree drawings. Comput. Geom.: Theor. Appl. 23(2), 153–162 (2002)
25.
Zurück zum Zitat C.C. Cheng, C.A. Duncan, M.T. Goodrich, S.G. Kobourov, Drawing planar graphs with circular arcs. Discr. Comput. Geom. 25(3), 405–418 (2001)MathSciNetMATHCrossRef C.C. Cheng, C.A. Duncan, M.T. Goodrich, S.G. Kobourov, Drawing planar graphs with circular arcs. Discr. Comput. Geom. 25(3), 405–418 (2001)MathSciNetMATHCrossRef
26.
Zurück zum Zitat N. Chiba, T. Yamanouchi, T. Nishizeki, Linear algorithms for convex drawings of planar graphs, in Progress in Graph Theory, ed. by J.A. Bondy, U.S.R. Murty (Academic Press, New York, 1984), pp. 153–173 N. Chiba, T. Yamanouchi, T. Nishizeki, Linear algorithms for convex drawings of planar graphs, in Progress in Graph Theory, ed. by J.A. Bondy, U.S.R. Murty (Academic Press, New York, 1984), pp. 153–173
27.
Zurück zum Zitat M. Chrobak, M.T. Goodrich, R. Tamassia, Convex drawings of graphs in two and three dimensions (preliminary version), in Symposium on Computational Geometry (SoCG’96), Philadelphia, Pennsylvania, USA, 1996, pp. 319–328 M. Chrobak, M.T. Goodrich, R. Tamassia, Convex drawings of graphs in two and three dimensions (preliminary version), in Symposium on Computational Geometry (SoCG’96), Philadelphia, Pennsylvania, USA, 1996, pp. 319–328
28.
Zurück zum Zitat M. Chrobak, G. Kant, Convex grid drawings of 3-connected planar graphs. Int. J. Comput. Geom. Appl. Limerick, Ireland, 7(3), 211–223 (1997) M. Chrobak, G. Kant, Convex grid drawings of 3-connected planar graphs. Int. J. Comput. Geom. Appl. Limerick, Ireland, 7(3), 211–223 (1997)
29.
Zurück zum Zitat M. Chrobak, T.H. Payne, A linear-time algorithm for drawing a planar graph on a grid. Inform. Process. Lett. 54(4), 241–246 (1995)MathSciNetMATHCrossRef M. Chrobak, T.H. Payne, A linear-time algorithm for drawing a planar graph on a grid. Inform. Process. Lett. 54(4), 241–246 (1995)MathSciNetMATHCrossRef
31.
Zurück zum Zitat P.F. Cortese, G. Di Battista, F. Frati, M. Patrignani, M. Pizzonia, C-planarity of c-connected clustered graphs. J. Graph Algorithm. Appl. Carleton University, Ottawa, Canada, 12(2), 225–262 (2008) P.F. Cortese, G. Di Battista, F. Frati, M. Patrignani, M. Pizzonia, C-planarity of c-connected clustered graphs. J. Graph Algorithm. Appl. Carleton University, Ottawa, Canada, 12(2), 225–262 (2008)
32.
Zurück zum Zitat P.F. Cortese, G. Di Battista, M. Patrignani, M. Pizzonia, Clustering cycles into cycles of clusters. J. Graph Algorithm. Appl. 9(3), 391–413 (2005)MATHCrossRef P.F. Cortese, G. Di Battista, M. Patrignani, M. Pizzonia, Clustering cycles into cycles of clusters. J. Graph Algorithm. Appl. 9(3), 391–413 (2005)MATHCrossRef
33.
Zurück zum Zitat P.F. Cortese, G. Di Battista, M. Patrignani, M. Pizzonia, On embedding a cycle in a plane graph. Discr. Math. 309(7), 1856–1869 (2009)MATHCrossRef P.F. Cortese, G. Di Battista, M. Patrignani, M. Pizzonia, On embedding a cycle in a plane graph. Discr. Math. 309(7), 1856–1869 (2009)MATHCrossRef
34.
Zurück zum Zitat P. Crescenzi, G. Di Battista, A. Piperno, A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom.: Theor. Appl. 2, 187–200 (1992) P. Crescenzi, G. Di Battista, A. Piperno, A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom.: Theor. Appl. 2, 187–200 (1992)
35.
Zurück zum Zitat E. Dahlhaus, A linear time algorithm to recognize clustered graphs and its parallelization, in Latin American Symposium on Theoretical Informatics (LATIN’98), ed. by C.L. Lucchesi, A.V. Moura. LNCS, Springer, vol. 1380 (1998), pp. 239–248 E. Dahlhaus, A linear time algorithm to recognize clustered graphs and its parallelization, in Latin American Symposium on Theoretical Informatics (LATIN’98), ed. by C.L. Lucchesi, A.V. Moura. LNCS, Springer, vol. 1380 (1998), pp. 239–248
36.
Zurück zum Zitat H. de Fraysseix, J. Pach, R. Pollack, Small sets supporting Fáry embeddings of planar graphs, in Symposium on Theory of Computing (STOC’88), ed. by J. Simon, Chicago, Illinois, USA, (1988), pp. 426–433 H. de Fraysseix, J. Pach, R. Pollack, Small sets supporting Fáry embeddings of planar graphs, in Symposium on Theory of Computing (STOC’88), ed. by J. Simon, Chicago, Illinois, USA, (1988), pp. 426–433
39.
Zurück zum Zitat G. Di Battista, G. Drovandi, F. Frati, How to draw a clustered tree. J. Discr. Algorithm. 7(4), 479–499 (2009)MATHCrossRef G. Di Battista, G. Drovandi, F. Frati, How to draw a clustered tree. J. Discr. Algorithm. 7(4), 479–499 (2009)MATHCrossRef
40.
Zurück zum Zitat G. Di Battista, P. Eades, R. Tamassia, I.G. Tollis, Graph Drawing (Prentice Hall, Upper Saddle River, NJ, 1999)MATH G. Di Battista, P. Eades, R. Tamassia, I.G. Tollis, Graph Drawing (Prentice Hall, Upper Saddle River, NJ, 1999)MATH
41.
Zurück zum Zitat G. Di Battista, F. Frati, Efficient c-planarity testing for embedded flat clustered graphs with small faces. J. Graph Algorithm. Appl. 13(3), 349–378 (2009)MATHCrossRef G. Di Battista, F. Frati, Efficient c-planarity testing for embedded flat clustered graphs with small faces. J. Graph Algorithm. Appl. 13(3), 349–378 (2009)MATHCrossRef
43.
Zurück zum Zitat G. Di Battista, W. Lenhart, G. Liotta, Proximity drawability: a survey, in Graph Drawing (GD’94), ed. by R. Tamassia, I.G. Tollis. LNCS, Princeton, New Jersey, USA, vol. 894 (1994), pp. 328–339 G. Di Battista, W. Lenhart, G. Liotta, Proximity drawability: a survey, in Graph Drawing (GD’94), ed. by R. Tamassia, I.G. Tollis. LNCS, Princeton, New Jersey, USA, vol. 894 (1994), pp. 328–339
44.
Zurück zum Zitat G. Di Battista, G. Liotta, S. Whitesides, The strength of weak proximity. J. Discr. Algorithm. 4(3), 384–400 (2006)MATHCrossRef G. Di Battista, G. Liotta, S. Whitesides, The strength of weak proximity. J. Discr. Algorithm. 4(3), 384–400 (2006)MATHCrossRef
45.
Zurück zum Zitat G. Di Battista, W.P. Liu, I. Rival, Bipartite graphs, upward drawings, and planarity. Inform. Process. Lett. 36(6), 317–322 (1990)MathSciNetMATHCrossRef G. Di Battista, W.P. Liu, I. Rival, Bipartite graphs, upward drawings, and planarity. Inform. Process. Lett. 36(6), 317–322 (1990)MathSciNetMATHCrossRef
46.
Zurück zum Zitat G. Di Battista, R. Tamassia, Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61, 175–198 (1988)MATHCrossRef G. Di Battista, R. Tamassia, Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61, 175–198 (1988)MATHCrossRef
47.
48.
Zurück zum Zitat G. Di Battista, R. Tamassia, I.G. Tollis, Area requirement and symmetry display of planar upward drawings. Discr. Comput. Geom. 7, 381–401 (1992)MATHCrossRef G. Di Battista, R. Tamassia, I.G. Tollis, Area requirement and symmetry display of planar upward drawings. Discr. Comput. Geom. 7, 381–401 (1992)MATHCrossRef
49.
Zurück zum Zitat G. Di Battista, R. Tamassia, L. Vismara, Incremental convex planarity testing. Inform. Comput. 169(1), 94–126 (2001)MATHCrossRef G. Di Battista, R. Tamassia, L. Vismara, Incremental convex planarity testing. Inform. Comput. 169(1), 94–126 (2001)MATHCrossRef
50.
Zurück zum Zitat W. Didimo, F. Giordano, G. Liotta, Upward spirality and upward planarity testing, in Graph Drawing (GD’05), ed. by P. Healy, N. Nikolov. LNCS, vol. 3843 (2005), pp. 117–128 W. Didimo, F. Giordano, G. Liotta, Upward spirality and upward planarity testing, in Graph Drawing (GD’05), ed. by P. Healy, N. Nikolov. LNCS, vol. 3843 (2005), pp. 117–128
52.
Zurück zum Zitat M.B. Dillencourt, W.D. Smith, Graph-theoretical conditions for inscribability and Delaunay realizability. Discr. Math. 161(1–3), 63–77 (1996)MathSciNetMATHCrossRef M.B. Dillencourt, W.D. Smith, Graph-theoretical conditions for inscribability and Delaunay realizability. Discr. Math. 161(1–3), 63–77 (1996)MathSciNetMATHCrossRef
53.
Zurück zum Zitat D. Dolev, H.W. Trickey, On linear area embedding of planar graphs. Technical report, Stanford University, Stanford, CA, 1981 D. Dolev, H.W. Trickey, On linear area embedding of planar graphs. Technical report, Stanford University, Stanford, CA, 1981
54.
Zurück zum Zitat P. Eades, Q. Feng, X. Lin, H. Nagamochi, Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)MathSciNetMATHCrossRef P. Eades, Q. Feng, X. Lin, H. Nagamochi, Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)MathSciNetMATHCrossRef
55.
Zurück zum Zitat P. Eades, Q. Feng, H. Nagamochi, Drawing clustered graphs on an orthogonal grid. J. Graph Algorithm. Appl. 3(4), 3–29 (1999)MathSciNetMATHCrossRef P. Eades, Q. Feng, H. Nagamochi, Drawing clustered graphs on an orthogonal grid. J. Graph Algorithm. Appl. 3(4), 3–29 (1999)MathSciNetMATHCrossRef
56.
Zurück zum Zitat P. Eades, S. Whitesides, The logic engine and the realization problem for nearest neighbor graphs. Theor. Comput. Sci. 169(1), 23–37 (1996)MathSciNetMATHCrossRef P. Eades, S. Whitesides, The logic engine and the realization problem for nearest neighbor graphs. Theor. Comput. Sci. 169(1), 23–37 (1996)MathSciNetMATHCrossRef
57.
Zurück zum Zitat P. Eades, S. Whitesides, The realization problem for Euclidean minimum spanning trees in NP-hard. Algorithmica 16(1), 60–82 (1996)MathSciNetMATHCrossRef P. Eades, S. Whitesides, The realization problem for Euclidean minimum spanning trees in NP-hard. Algorithmica 16(1), 60–82 (1996)MathSciNetMATHCrossRef
58.
Zurück zum Zitat D. Eppstein, M.T. Goodrich, Succinct greedy graph drawing in the hyperbolic plane, in Graph Drawing (GD’08), ed. by I.G. Tollis, M. Patrignani. LNCS, vol. 5417 (2009), pp. 14–25 D. Eppstein, M.T. Goodrich, Succinct greedy graph drawing in the hyperbolic plane, in Graph Drawing (GD’08), ed. by I.G. Tollis, M. Patrignani. LNCS, vol. 5417 (2009), pp. 14–25
59.
Zurück zum Zitat I. Fáry, On straight line representation of planar graphs. Acta Sci. Math. 11, 229–233 (1948)MATH I. Fáry, On straight line representation of planar graphs. Acta Sci. Math. 11, 229–233 (1948)MATH
60.
Zurück zum Zitat S. Felsner, G. Liotta, S.K. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithm. Appl. 7(4), 363–398 (2003)MathSciNetMATHCrossRef S. Felsner, G. Liotta, S.K. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithm. Appl. 7(4), 363–398 (2003)MathSciNetMATHCrossRef
61.
Zurück zum Zitat Q. Feng, Algorithms for drawing clustered graphs. Ph.D. thesis, The University of Newcastle, Australia, 1997 Q. Feng, Algorithms for drawing clustered graphs. Ph.D. thesis, The University of Newcastle, Australia, 1997
62.
Zurück zum Zitat Q. Feng, R.F. Cohen, P. Eades, How to draw a planar clustered graph, in Computing and Combinatorics (COCOON’95), ed. by D. Du, M. Li. LNCS, Xi’an, China, vol. 959 (1995), pp. 21–30 Q. Feng, R.F. Cohen, P. Eades, How to draw a planar clustered graph, in Computing and Combinatorics (COCOON’95), ed. by D. Du, M. Li. LNCS, Xi’an, China, vol. 959 (1995), pp. 21–30
63.
Zurück zum Zitat Q. Feng, R.F. Cohen, P. Eades, Planarity for clustered graphs, in European Symposium on Algorithms (ESA’95), ed. by P.G. Spirakis. LNCS, Corfu, Greece, vol. 979 (1995), pp. 213–226 Q. Feng, R.F. Cohen, P. Eades, Planarity for clustered graphs, in European Symposium on Algorithms (ESA’95), ed. by P.G. Spirakis. LNCS, Corfu, Greece, vol. 979 (1995), pp. 213–226
64.
Zurück zum Zitat F. Frati, Straight-line drawings of outerplanar graphs in O(dnlogn) area, in Canadian Conference on Computational Geometry (CCCG’07), Carleton University, Ottawa, Canada 2007, pp. 225–228 F. Frati, Straight-line drawings of outerplanar graphs in O(dnlogn) area, in Canadian Conference on Computational Geometry (CCCG’07), Carleton University, Ottawa, Canada 2007, pp. 225–228
65.
Zurück zum Zitat F. Frati, On minimum area planar upward drawings of directed trees and other families of directed acyclic graphs. Int. J. Comput. Geom. Appl. 18(3), 251–271 (2008)MathSciNetMATHCrossRef F. Frati, On minimum area planar upward drawings of directed trees and other families of directed acyclic graphs. Int. J. Comput. Geom. Appl. 18(3), 251–271 (2008)MathSciNetMATHCrossRef
66.
Zurück zum Zitat F. Frati, Straight-line orthogonal drawings of binary and ternary trees, in Graph Drawing (GD’07), ed. by S.-H. Hong, T. Nishizeki, W. Quan. LNCS, Sydney, Australia, vol. 4875 (2008), pp. 76–87 F. Frati, Straight-line orthogonal drawings of binary and ternary trees, in Graph Drawing (GD’07), ed. by S.-H. Hong, T. Nishizeki, W. Quan. LNCS, Sydney, Australia, vol. 4875 (2008), pp. 76–87
67.
Zurück zum Zitat F. Frati, Lower bounds on the area requirements of series-parallel graphs. Discr. Math. Theor. Comput. Sci. 12(5), 139–174 (2010)MathSciNet F. Frati, Lower bounds on the area requirements of series-parallel graphs. Discr. Math. Theor. Comput. Sci. 12(5), 139–174 (2010)MathSciNet
68.
Zurück zum Zitat F. Frati, M. Kaufmann, Polynomial area bounds for MST embeddings of trees. Tech. report RT-DIA-122-2008, Dept. of Computer Science and Automation, Roma Tre University, Rome, 2008 F. Frati, M. Kaufmann, Polynomial area bounds for MST embeddings of trees. Tech. report RT-DIA-122-2008, Dept. of Computer Science and Automation, Roma Tre University, Rome, 2008
69.
Zurück zum Zitat F. Frati, M. Patrignani, A note on minimum area straight-line drawings of planar graphs, in Graph Drawing (GD’07), ed. by S.H. Hong, T. Nishizeki. LNCS, Sydney, Australia, vol. 4875 (2007), pp. 339–344 F. Frati, M. Patrignani, A note on minimum area straight-line drawings of planar graphs, in Graph Drawing (GD’07), ed. by S.H. Hong, T. Nishizeki. LNCS, Sydney, Australia, vol. 4875 (2007), pp. 339–344
70.
Zurück zum Zitat A. Garg, M.T. Goodrich, R. Tamassia, Planar upward tree drawings with optimal area. Int. J. Comput. Geom. Appl. 6(3), 333–356 (1996)MathSciNetMATHCrossRef A. Garg, M.T. Goodrich, R. Tamassia, Planar upward tree drawings with optimal area. Int. J. Comput. Geom. Appl. 6(3), 333–356 (1996)MathSciNetMATHCrossRef
71.
Zurück zum Zitat A. Garg, A. Rusu, Area-efficient order-preserving planar straight-line drawings of ordered trees. Int. J. Comput. Geom. Appl. 13(6), 487–505 (2003)MathSciNetMATHCrossRef A. Garg, A. Rusu, Area-efficient order-preserving planar straight-line drawings of ordered trees. Int. J. Comput. Geom. Appl. 13(6), 487–505 (2003)MathSciNetMATHCrossRef
72.
Zurück zum Zitat A. Garg, A. Rusu, Straight-line drawings of general trees with linear area and arbitrary aspect ratio, in Computational Science and Its Applications (ICCSA’03), ed. by V. Kumar, M.L. Gavrilova, C.J.K. Tan, P. L’Ecuyer. LNCS, Montreal, Canada, vol. 2669 (2003), pp. 876–885 A. Garg, A. Rusu, Straight-line drawings of general trees with linear area and arbitrary aspect ratio, in Computational Science and Its Applications (ICCSA’03), ed. by V. Kumar, M.L. Gavrilova, C.J.K. Tan, P. L’Ecuyer. LNCS, Montreal, Canada, vol. 2669 (2003), pp. 876–885
73.
Zurück zum Zitat A. Garg, A. Rusu, Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. J. Graph Algorithm. Appl. Rome, Italy, 8(2), 135–160 (2004) A. Garg, A. Rusu, Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. J. Graph Algorithm. Appl. Rome, Italy, 8(2), 135–160 (2004)
74.
Zurück zum Zitat A. Garg, A. Rusu, Area-efficient planar straight-line drawings of outerplanar graphs. Discr. Appl. Math. 155(9), 1116–1140 (2007)MathSciNetMATHCrossRef A. Garg, A. Rusu, Area-efficient planar straight-line drawings of outerplanar graphs. Discr. Appl. Math. 155(9), 1116–1140 (2007)MathSciNetMATHCrossRef
75.
Zurück zum Zitat A. Garg, R. Tamassia, Efficient computation of planar straight-line upward drawings, in Graph Drawing (Proc. ALCOM Workshop on Graph Drawing), Sevres, Paris, 1994, pp. 298–306 A. Garg, R. Tamassia, Efficient computation of planar straight-line upward drawings, in Graph Drawing (Proc. ALCOM Workshop on Graph Drawing), Sevres, Paris, 1994, pp. 298–306
76.
Zurück zum Zitat A. Garg, R. Tamassia, On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)MathSciNetMATHCrossRef A. Garg, R. Tamassia, On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)MathSciNetMATHCrossRef
77.
Zurück zum Zitat M.T. Goodrich, G.S. Lueker, J.Z. Sun, C-planarity of extrovert clustered graphs, in Graph Drawing (GD’05), ed. by P. Healy, N. Nikolov. LNCS, Limerick, Ireland, vol. 3843 (2005), pp. 211–222 M.T. Goodrich, G.S. Lueker, J.Z. Sun, C-planarity of extrovert clustered graphs, in Graph Drawing (GD’05), ed. by P. Healy, N. Nikolov. LNCS, Limerick, Ireland, vol. 3843 (2005), pp. 211–222
78.
Zurück zum Zitat M.T. Goodrich, D. Strash, Succinct greedy geometric routing in the Euclidean plane, in Algorithms and Computation (ISAAC’09), ed. by Y. Dong, D.-Z. Du, O.H. Ibarra. LNCS, Hawai, USA, vol. 5878 (2009), pp. 781–791 M.T. Goodrich, D. Strash, Succinct greedy geometric routing in the Euclidean plane, in Algorithms and Computation (ISAAC’09), ed. by Y. Dong, D.-Z. Du, O.H. Ibarra. LNCS, Hawai, USA, vol. 5878 (2009), pp. 781–791
79.
Zurück zum Zitat M.T. Goodrich, C.G. Wagner, A framework for drawing planar graphs with curves and polylines. J. Algorithms 37(2), 399–421 (2000)MathSciNetMATHCrossRef M.T. Goodrich, C.G. Wagner, A framework for drawing planar graphs with curves and polylines. J. Algorithms 37(2), 399–421 (2000)MathSciNetMATHCrossRef
80.
Zurück zum Zitat C. Gutwenger, M. Jünger, S. Leipert, P. Mutzel, M. Percan, R. Weiskircher, Advances in c-planarity testing of clustered graphs, in Graph Drawing (GD’02), ed. by S.G. Kobourov, M.T. Goodrich. LNCS, Irvine, California, USA, vol. 2528 (2002), pp. 220–235 C. Gutwenger, M. Jünger, S. Leipert, P. Mutzel, M. Percan, R. Weiskircher, Advances in c-planarity testing of clustered graphs, in Graph Drawing (GD’02), ed. by S.G. Kobourov, M.T. Goodrich. LNCS, Irvine, California, USA, vol. 2528 (2002), pp. 220–235
81.
Zurück zum Zitat C. Gutwenger, P. Mutzel, Planar polyline drawings with good angular resolution, in Graph Drawing (GD’98), ed. by S. Whitesides. LNCS, Montréal, Canada, vol. 1547 (1998), pp. 167–182 C. Gutwenger, P. Mutzel, Planar polyline drawings with good angular resolution, in Graph Drawing (GD’98), ed. by S. Whitesides. LNCS, Montréal, Canada, vol. 1547 (1998), pp. 167–182
82.
Zurück zum Zitat X. He, Grid embedding of 4-connected plane graphs. Discr. Comput. Geom. 17(3), 339–358 (1997)MATHCrossRef X. He, Grid embedding of 4-connected plane graphs. Discr. Comput. Geom. 17(3), 339–358 (1997)MATHCrossRef
83.
Zurück zum Zitat X. He, H. Zhang, Succinct convex greedy drawing of 3-connected plane graphs, in Symposium on Discrete Algorithms (SODA’11), San Francisco, California, USA, 2011, pp. 1477–1486 X. He, H. Zhang, Succinct convex greedy drawing of 3-connected plane graphs, in Symposium on Discrete Algorithms (SODA’11), San Francisco, California, USA, 2011, pp. 1477–1486
84.
85.
Zurück zum Zitat V. Jelínek, E. Jelínková, J. Kratochvíl, B. Lidický, Clustered planarity: embedded clustered graphs with two-component clusters, in Graph Drawing (GD’08), ed. by I.G. Tollis, M. Patrignani. LNCS, Heraklion, Crete, Greece, vol. 5417 (2008), pp. 121–132 V. Jelínek, E. Jelínková, J. Kratochvíl, B. Lidický, Clustered planarity: embedded clustered graphs with two-component clusters, in Graph Drawing (GD’08), ed. by I.G. Tollis, M. Patrignani. LNCS, Heraklion, Crete, Greece, vol. 5417 (2008), pp. 121–132
86.
Zurück zum Zitat V. Jelínek, O. Suchý, M. Tesař, T. Vyskočil, Clustered planarity: clusters with few outgoing edges, in Graph Drawing (GD’08), ed. by I.G. Tollis, M. Patrignani. LNCS, Heraklion, Crete, Greece, vol. 5417 (2008), pp. 102–113 V. Jelínek, O. Suchý, M. Tesař, T. Vyskočil, Clustered planarity: clusters with few outgoing edges, in Graph Drawing (GD’08), ed. by I.G. Tollis, M. Patrignani. LNCS, Heraklion, Crete, Greece, vol. 5417 (2008), pp. 102–113
87.
Zurück zum Zitat E. Jelínková, J. Kára, J. Kratochvíl, M. Pergel, O. Suchý, T. Vyskočil, Clustered planarity: small clusters in cycles and Eulerian graphs. J. Graph Algorithm Appl. 13(3), 379–422 (2009)MATHCrossRef E. Jelínková, J. Kára, J. Kratochvíl, M. Pergel, O. Suchý, T. Vyskočil, Clustered planarity: small clusters in cycles and Eulerian graphs. J. Graph Algorithm Appl. 13(3), 379–422 (2009)MATHCrossRef
89.
Zurück zum Zitat G. Kant, X. He, Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci. 172(1–2), 175–193 (1997)MathSciNetMATHCrossRef G. Kant, X. He, Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci. 172(1–2), 175–193 (1997)MathSciNetMATHCrossRef
90.
Zurück zum Zitat M. Kaufmann, Polynomial area bounds for MST embeddings of trees, in Graph Drawing (GD’07), ed. by S.H. Hong, T. Nishizeki, W. Quan. LNCS, Sydney, Australia, vol. 4875 (2007), pp. 88–100 M. Kaufmann, Polynomial area bounds for MST embeddings of trees, in Graph Drawing (GD’07), ed. by S.H. Hong, T. Nishizeki, W. Quan. LNCS, Sydney, Australia, vol. 4875 (2007), pp. 88–100
91.
Zurück zum Zitat M. Kaufmann, D. Wagner (eds.), Drawing Graphs, Methods and Models. Lecture Notes in Computer Science (Springer-Verlag, New York/Heidelberg, 2001) M. Kaufmann, D. Wagner (eds.), Drawing Graphs, Methods and Models. Lecture Notes in Computer Science (Springer-Verlag, New York/Heidelberg, 2001)
92.
Zurück zum Zitat S.K. Kim, Simple algorithms for orthogonal upward drawings of binary and ternary trees, in Canadian Conference on Computational Geometry (CCCG’95), Quebec City, Quebec, Canada, 1995, pp. 115–120 S.K. Kim, Simple algorithms for orthogonal upward drawings of binary and ternary trees, in Canadian Conference on Computational Geometry (CCCG’95), Quebec City, Quebec, Canada, 1995, pp. 115–120
93.
Zurück zum Zitat R. Kleinberg, Geographic routing using hyperbolic space, in IEEE Conference on Computer Communications (INFOCOM’07), Anchorage, Alaska, USA, 2007, pp. 1902–1909 R. Kleinberg, Geographic routing using hyperbolic space, in IEEE Conference on Computer Communications (INFOCOM’07), Anchorage, Alaska, USA, 2007, pp. 1902–1909
94.
95.
Zurück zum Zitat W. Lenhart, G. Liotta, Proximity drawings of outerplanar graphs, in Graph Drawing (GD’96), ed. by S.C. North. LNCS, Berkeley, California, USA, vol. 1190 (1996), pp. 286–302 W. Lenhart, G. Liotta, Proximity drawings of outerplanar graphs, in Graph Drawing (GD’96), ed. by S.C. North. LNCS, Berkeley, California, USA, vol. 1190 (1996), pp. 286–302
96.
Zurück zum Zitat G. Liotta, Computing proximity drawings of graphs. Ph.D. thesis, University of Rome “La Sapienza.” Rome, 1995 G. Liotta, Computing proximity drawings of graphs. Ph.D. thesis, University of Rome “La Sapienza.” Rome, 1995
97.
Zurück zum Zitat G. Liotta, R. Tamassia, I.G. Tollis, P. Vocca, Area requirement of Gabriel drawings, in Algorithms and Complexity (CIAC’97), ed. by G.C. Bongiovanni, D.P. Bovet, G. Di Battista. LNCS, Rome, Italy, vol. 1203 (1997), pp. 135–146 G. Liotta, R. Tamassia, I.G. Tollis, P. Vocca, Area requirement of Gabriel drawings, in Algorithms and Complexity (CIAC’97), ed. by G.C. Bongiovanni, D.P. Bovet, G. Di Battista. LNCS, Rome, Italy, vol. 1203 (1997), pp. 135–146
98.
Zurück zum Zitat A. Lubiw, N. Sleumer, Maximal outerplanar graphs are relative neighborhood graphs, in Canadian Conference on Computational Geometry (CCCG’93), Waterloo, Ontario, Canada, 1993, pp. 198–203 A. Lubiw, N. Sleumer, Maximal outerplanar graphs are relative neighborhood graphs, in Canadian Conference on Computational Geometry (CCCG’93), Waterloo, Ontario, Canada, 1993, pp. 198–203
99.
100.
Zurück zum Zitat D. Mondal, R.I. Nishat, Md. S. Rahman, Md. J. Alam, Minimum-area drawings of plane 3-trees, in Canadian Conference on Computational Geometry (CCCG’10), Manitoba, Canada, 2010, pp. 191–194 D. Mondal, R.I. Nishat, Md. S. Rahman, Md. J. Alam, Minimum-area drawings of plane 3-trees, in Canadian Conference on Computational Geometry (CCCG’10), Manitoba, Canada, 2010, pp. 191–194
101.
102.
Zurück zum Zitat T. Nishizeki, N. Chiba, Planar Graphs: Theory and Algorithms (North Holland, Amsterdam, 1988)MATH T. Nishizeki, N. Chiba, Planar Graphs: Theory and Algorithms (North Holland, Amsterdam, 1988)MATH
103.
104.
Zurück zum Zitat A. Papakostas, Upward planarity testing of outerplanar dags, in Graph Drawing, ed. by R. Tamassia, I.G. Tollis. LNCS, Springer, vol. 894 (1994), pp. 298–306 A. Papakostas, Upward planarity testing of outerplanar dags, in Graph Drawing, ed. by R. Tamassia, I.G. Tollis. LNCS, Springer, vol. 894 (1994), pp. 298–306
105.
Zurück zum Zitat P. Penna, P. Vocca, Proximity drawings in polynomial area and volume. Comput. Geom.: Theor. Appl. 29(2), 91–116 (2004) P. Penna, P. Vocca, Proximity drawings in polynomial area and volume. Comput. Geom.: Theor. Appl. 29(2), 91–116 (2004)
106.
Zurück zum Zitat S. Rabinowitz, O(n 3) bounds for the area of a convex lattice n-gon. Geombinatorics 2, 85–88 (1993)MathSciNetMATH S. Rabinowitz, O(n 3) bounds for the area of a convex lattice n-gon. Geombinatorics 2, 85–88 (1993)MathSciNetMATH
107.
Zurück zum Zitat A. Rao, C.H. Papadimitriou, S. Shenker, I. Stoica, Geographic routing without location information, in ACM International Conference on Mobile Computing and Networking (MOBICOM’03), ed. by D.B. Johnson, A.D. Joseph, N.H. Vaidya, San Diego, California, USA, 2003, pp. 96–108 A. Rao, C.H. Papadimitriou, S. Shenker, I. Stoica, Geographic routing without location information, in ACM International Conference on Mobile Computing and Networking (MOBICOM’03), ed. by D.B. Johnson, A.D. Joseph, N.H. Vaidya, San Diego, California, USA, 2003, pp. 96–108
108.
Zurück zum Zitat R.C. Read, A new method for drawing a graph given the cyclic order of the edges at each vertex. Congressus Numerantium 56, 31–44 (1987)MathSciNet R.C. Read, A new method for drawing a graph given the cyclic order of the edges at each vertex. Congressus Numerantium 56, 31–44 (1987)MathSciNet
109.
Zurück zum Zitat G. Rote, Strictly convex drawings of planar graphs, in Symposium on Discrete Algorithms (SODA’05), Vancouver, BC, Canada, 2005, pp. 728–734 G. Rote, Strictly convex drawings of planar graphs, in Symposium on Discrete Algorithms (SODA’05), Vancouver, BC, Canada, 2005, pp. 728–734
110.
Zurück zum Zitat W. Schnyder, Embedding planar graphs on the grid, in Symposium on Discrete Algorithms (SODA’90), San Francisco, California, USA, 1990, pp. 138–148 W. Schnyder, Embedding planar graphs on the grid, in Symposium on Discrete Algorithms (SODA’90), San Francisco, California, USA, 1990, pp. 138–148
111.
Zurück zum Zitat W. Schnyder, W. Trotter, Convex drawings of planar graphs. Abstr. Am. Math. Soc. 92T-05-135 (1992) W. Schnyder, W. Trotter, Convex drawings of planar graphs. Abstr. Am. Math. Soc. 92T-05-135 (1992)
112.
Zurück zum Zitat C.S. Shin, S.K. Kim, K.Y. Chwa, Area-efficient algorithms for straight-line tree drawings. Comput. Geom.: Theor. Appl. 15(4), 175–202 (2000) C.S. Shin, S.K. Kim, K.Y. Chwa, Area-efficient algorithms for straight-line tree drawings. Comput. Geom.: Theor. Appl. 15(4), 175–202 (2000)
113.
Zurück zum Zitat S.K. Stein, Convex maps. Am. Math. Soc. 2, 464–466 (1951)MATH S.K. Stein, Convex maps. Am. Math. Soc. 2, 464–466 (1951)MATH
114.
Zurück zum Zitat M.A. Stern, Ueber eine zahlentheoretische funktion. J. Reine Angew. Math. 55, 193–220 (1858)MATHCrossRef M.A. Stern, Ueber eine zahlentheoretische funktion. J. Reine Angew. Math. 55, 193–220 (1858)MATHCrossRef
115.
Zurück zum Zitat R. Tamassia (ed.), Handbook of Graph Drawing and Visualization (CRC Press), to appear R. Tamassia (ed.), Handbook of Graph Drawing and Visualization (CRC Press), to appear
116.
Zurück zum Zitat R. Tamassia, I.G. Tollis, A unified approach to visibility representation of planar graphs. Discr. Comput. Geom. 1, 321–341 (1986)MathSciNetMATHCrossRef R. Tamassia, I.G. Tollis, A unified approach to visibility representation of planar graphs. Discr. Comput. Geom. 1, 321–341 (1986)MathSciNetMATHCrossRef
117.
Zurück zum Zitat C. Thomassen, Planarity and duality of finite and infinite graphs. J. Comb. Theor. Ser. B 29(2), 244–271 (1980)MATHCrossRef C. Thomassen, Planarity and duality of finite and infinite graphs. J. Comb. Theor. Ser. B 29(2), 244–271 (1980)MATHCrossRef
118.
Zurück zum Zitat C. Thomassen, Plane representations of graphs, in Progress in Graph Theory (Academic, 1984), pp. 43–69 C. Thomassen, Plane representations of graphs, in Progress in Graph Theory (Academic, 1984), pp. 43–69
122.
Zurück zum Zitat K. Wagner, Bemerkungen zum vierfarbenproblem. Jahresbericht. German. Math.-Verein 2, 26–32 (1936) K. Wagner, Bemerkungen zum vierfarbenproblem. Jahresbericht. German. Math.-Verein 2, 26–32 (1936)
123.
Zurück zum Zitat D. Woods, Drawing planar graphs. Ph.D. thesis, Stanford University, Stanford, CA, 1982 D. Woods, Drawing planar graphs. Ph.D. thesis, Stanford University, Stanford, CA, 1982
125.
Zurück zum Zitat H. Zhang, S. Sadasivam, On planar polyline drawings, in Graph Drawing, ed. by S.-H. Hong, T. Nishizeki, W. Quan. LNCS, Springer, vol. 4875 (2008), pp. 213–218 H. Zhang, S. Sadasivam, On planar polyline drawings, in Graph Drawing, ed. by S.-H. Hong, T. Nishizeki, W. Quan. LNCS, Springer, vol. 4875 (2008), pp. 213–218
126.
Zurück zum Zitat X. Zhou, T. Hikino, T. Nishizeki, Small grid drawings of planar graphs with balanced bipartition, in Algorithms and Computation (WALCOM’10), ed. by Md. S. Rahman, S. Fujita. LNCS, Dhaka, Bangladesh, vol. 5942 (2010), pp. 47–57 X. Zhou, T. Hikino, T. Nishizeki, Small grid drawings of planar graphs with balanced bipartition, in Algorithms and Computation (WALCOM’10), ed. by Md. S. Rahman, S. Fujita. LNCS, Dhaka, Bangladesh, vol. 5942 (2010), pp. 47–57
Metadaten
Titel
Drawing Trees, Outerplanar Graphs, Series-Parallel Graphs, and Planar Graphs in a Small Area
verfasst von
Giuseppe Di Battista
Fabrizio Frati
Copyright-Jahr
2013
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-0110-0_9