Abstract
In this paper, we introduce and study the multilevel-planarity testing problem, which is a generalization of upward planarity and level planarity. Let \(G = (V, E)\) be a directed graph and let \(\ell : V \rightarrow \mathcal P(\mathbb Z)\) be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of G is a planar drawing of G such that the y-coordinate of each vertex \(v \in V\) is \(y(v) \in \ell (v)\), and each edge is drawn as a strictly y-monotone curve.
We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source and of oriented cycles. Complementing these algorithmic results, we show that multilevel-planarity testing is NP-complete even in very restricted cases.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Beyond level planarity. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 482–495. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-50106-2_37
Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Roselli, V.: The importance of being proper (in clustered-level planarity and \(T\)-level planarity). Theoretical Comput. Sci. 571, 1–9 (2015)
Angelini, P., et al.: Testing planarity of partially embedded graphs. ACM Trans. Alg. 11(4), 32:1–32:42 (2015)
Bachmaier, C., Brandenburg, F.J., Forster, M.: Radial level planarity testing and embedding in linear time. J. Graph Alg. Appl. 9(1), 53–97 (2005)
Barth, L., Brückner, G., Jungeblut, P., Radermacher, M.: Multilevel planarity (2018). https://arxiv.org/abs/1810.13297
Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)
Bertolazzi, P., Di Battista, G., Mannino, C., Tamassia, R.: Optimal upward planarity testing of single-source digraphs. SIAM J. Comput. 27(1), 132–169 (1998)
Brückner, G., Rutter, I.: Partial and constrained level planarity. In: Klein, P.N. (ed.) SODA 2017, pp. 2000–2011 (2017)
De Berg, M., Khosravi, A.: Optimal binary space partitions for segments in the plane. Int. J. Comput. Geom. Appl. 22(3), 187–205 (2012)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs, 1st edn. Prentice Hall PTR (1998)
Di Battista, G., Frati, F.: Efficient C-planarity testing for embedded flat clustered graphs with small faces. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 291–302. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77537-9_29
Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theoret. Comput. Sci. 61(2), 175–198 (1988)
Forster, M., Bachmaier, C.: Clustered level planarity. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 218–228. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24618-3_18
Garey, M.R., Johnson, D.S.: Two-processor scheduling with start-times and deadlines. SIAM J. Comput. 6(3), 416–426 (1977)
Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2002)
Harrigan, M., Healy, P.: Practical level planarity testing and layout with embedding constraints. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 62–68. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77537-9_9
Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. Theory Appl. 46(4), 466–492 (2013)
Jelínková, E., Kára, J., Kratochvíl, J., Pergel, M., Suchý, O., Vyskočil, T.: Clustered planarity: small clusters in Cycles and Eulerian Graphs. J. Graph Alg. Appl. 13(3), 379–422 (2009)
Jünger, M., Leipert, S.: Level planar embedding in linear time. In: Kratochvíyl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 72–81. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-46648-7_7
Klemz, B., Rote, G.: Ordered level planarity, geodesic planarity and bi-monotonicity. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 440–453. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73915-1_34
Leipert, S.: Level planarity testing and embedding in linear time. Ph.D. thesis, University of Cologne (1998)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Barth, L., Brückner, G., Jungeblut, P., Radermacher, M. (2019). Multilevel Planarity. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-10564-8_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-10563-1
Online ISBN: 978-3-030-10564-8
eBook Packages: Computer ScienceComputer Science (R0)