Abstract
An upward embedding of an embedded planar graph states, for each vertex v, which edges are incident to v “above” or “below” and, in turn, induces an upward orientation of the edges. In this paper we characterize the set of all upward embeddings and orientations of a plane graph by using a simple flow model. We take advantage of such a flow model to compute upward orientations with the minimum number of sources and sinks of 1-connected graphs. Our theoretical results allow us to easily compute visibility representations of 1-connected graphs while having a certain control over the width and the height of the computed drawings, and to deal with partial assignments of the upward embeddings “underlying” the visibility representations.
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References
R. K. Ahuja, T. L. Magnanti, and J. B. Orlin. Network flows. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management, pages 211–360. North-Holland, 1990.
P. Bertolazzi, G. Di Battista, and W. Didimo. Computing orthogonal drawings with the minimum numbr of bends. IEEE Transactions on Computers, 49(8), 2000.
P. Bertolazzi, G. Di Battista, G. Liotta, and C. Mannino. Upward drawings of triconnected digraphs. Algorithmica, 6(12):476–497, 1994.
P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. SIAM J. Comput., 27(1):13–169, 1998.
M. Bousset. A flow model of low complexity for twisting a layout. In Workshop of GD’93, pages 43–44, Paris, 1993.
J. Czyzowicz, A. Pelc, and I. Rival. Drawing orders with few slopes. Technical Report TR-87-12, Department of Computer Science, University of Ottawa, 1987.
H. de Fraysseix, P. O. de Mendez, and P. Rosenstiehl. Bipolar orientations revisited. Discrete Appl. Math., 56:157–179, 1995.
G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing. Prentice Hall, Upper Saddle River, NJ, 1999.
W. Didimo and M. Pizzonia. Upward embeddings and orientations of undirected planar graphs. Technical Report RT-DIA-65-2001, University of Roma Tre, 2001.
S. Fialko and P. Mutzel. A new approximation algorithm for the planar augmentation problem. In Symposium on Discrete Algorithms (SODA’ 98), pages 260–269, 1998.
A. Garg and R. Tamassia. On the computational complexity of upward and rectilinear planarity testing. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD’ 94), volume 894 of Lecture Notes Comput. Sci., pages 286–297. Springer-Verlag, 1995.
A. Garg and R. Tamassia. A new minimum cost flow algorithm with applications to graph drawing. In S. C. North, editor, Graph Drawing (Proc. GD’ 96), volume 1190 of Lecture Notes Comput. Sci., pages 201–216. Springer-Verlag, 1997.
G. Kant and H. L. Bodlaender. Planar graph augmentation problems. In Proc. 2nd Workshop Algorithms Data Struct., volume 519 of Lecture Notes Comput. Sci., pages 286–298. Springer-Verlag, 1991.
D. Kelly. Fundamentals of planar ordered sets. Discrete Math., 63:197–216, 1987.
D. Kelly and I. Rival. Planar lattices. Canad. J. Math., 27(3):636–665, 1975.
D. G. Kirkpatrick and S. K. Wismath. Weighted visibility graphs of bars and related flow problems. In Proc. 1st Workshop Algorithms Data Struct., volume 382 of Lecture Notes Comput. Sci., pages 325–334. Springer-Verlag, 1989.
M. Pizzonia. Engineering of Graph Drawing Algorithms for Applications. PhD thesis, Dipartimento di Informatica e Sistemistica, Università “La Sapienza” di Roma, 2001.
I. Rival. Reading, drawing, and order. In I. G. Rosenberg and G. Sabidussi, editors, Algebras and Orders, pages 359–404. Kluwer Academic Publishers, 1993.
R. Tamassia and I. G. Tollis. A unified approach to visibility representations of planar graphs. Discrete Comput. Geom., 1(4):321–341, 1986.
S. K. Wismath. Bar-Representable Visibility Graphs and Related Flow Problems. Ph.D. thesis, Dept. Comput. Sci., Univ. British Columbia, 1989.
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Didimo, W., Pizzonia, M. (2001). Upward Embeddings and Orientations of Undirected Planar Graphs. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_32
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DOI: https://doi.org/10.1007/3-540-44634-6_32
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