Abstract
In this paper we consider graphs whose edges are associated with a degree of importance, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts in which the readability of an edge is proportional to its importance, that is, more important edges have fewer crossings. We formalize this problem and study the case in which there exist three different degrees of importance. We give a polynomial-time testing algorithm when the graph induced by the two most important sets of edges is biconnected. We also discuss interesting relationships with other constrained-planarity problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Angelini, P., Bekos, M.A.: Hierarchical partial planarity. CoRR, 1707.06844, (2017). http://arxiv.org/abs/1707.06844
Angelini, P., Binucci, C., Da Lozzo, G., Didimo, W., Grilli, L., Montecchiani, F., Patrignani, M., Tollis, I.G.: Algorithms and bounds for drawing non-planar graphs with crossing-free subgraphs. Comput. Geom. 50, 34–48 (2015). doi:10.1016/j.comgeo.2015.07.002
Angelini, P., et al.: Simultaneous orthogonal planarity. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 532–545. Springer, Cham (2016). doi:10.1007/978-3-319-50106-2_41
Angelini, P., Da Lozzo, G., Neuwirth, D.: Advancements on SEFE and partitioned book embedding problems. Theor. Comput. Sci. 575, 71–89 (2015). doi:10.1016/j.tcs.2014.11.016
Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32 (2015). doi:10.1145/2629341
Angelini, P., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph. J. Discrete Algorithms 14, 150–172 (2012). doi:10.1016/j.jda.2011.12.015.
Bekos, M.A., van Dijk, T.C., Kindermann, P., Wolff, A.: Simultaneous drawing of planar graphs with right-angle crossings and few bends. J. Graph Algorithms Appl. 20(1), 133–158 (2016). doi:10.7155/jgaa.00388
Binucci, C., Brandes, U., Di Battista, G., Didimo, W., Gaertler, M., Palladino, P., Patrignani, M., Symvonis, A., Zweig, K.A.: Drawing trees in a streaming model. Inf. Process. Lett. 112(11), 418–422 (2012). doi:10.1016/j.ipl.2012.02.011
Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization, pp. 349–381. Chapman and Hall/CRC, London (2013)
Bläsius, T., Rutter, I.: Disconnectivity and relative positions in simultaneous embeddings. Comput. Geom. 48(6), 459–478 (2015). doi:10.1016/j.comgeo.2015.02.002
Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. ACM Trans. Algorithms 12(2), 16:1–16:46 (2016). doi:10.1145/2738054.
Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. 36(2), 117–130 (2007). doi:10.1016/j.comgeo.2006.05.006
Chan, T.M., Frati, F., Gutwenger, C., Lubiw, A., Mutzel, P., Schaefer, M.: Drawing partially embedded and simultaneously planar graphs. J. Graph Algorithms Appl. 19(2), 681–706 (2015). doi:10.7155/jgaa.00375
Da Lozzo, G., Rutter, I.: Planarity of streamed graphs. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 153–166. Springer, Cham (2015). doi:10.1007/978-3-319-18173-8_11
Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996). doi:10.1007/BF01961541
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996). doi:10.1137/S0097539794280736
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Wismath, S.K.: Planar and quasi-planar simultaneous geometric embedding. Comput. J. 58(11), 3126–3140 (2015). doi:10.1093/comjnl/bxv048
Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. Graph Algorithms Appl. 9(3), 347–364 (2005). doi:10.7155/jgaa.00113
Goodrich, M.T., Pszona, P.: Streamed graph drawing and the file maintenance problem. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 256–267. Springer, Cham (2013). doi:10.1007/978-3-319-03841-4_23
Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001). doi:10.1007/3-540-44541-2_8
Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. 46(4), 466–492 (2013). doi:10.1016/j.comgeo.2012.07.005
Kaufmann, M., Wagner, D. (eds.): Drawing Graphs, Methods and Models. LNCS, vol. 2025. Springer, Heidelberg (2001). doi:10.1016/0095-8956(91)90090-7
Kratochvíl, J.: String graphs. I. The number of critical nonstring graphs is infinite. J. Comb. Theory, Ser. B 52(1), 53–66 (1991). doi:10.1016/0095-8956(91)90090-7
Schaefer, M.: Picking planar edges; or, drawing a graph with a planar subgraph. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 13–24. Springer, Heidelberg (2014). doi:10.1007/978-3-662-45803-7_2
Tamassia, R., Liotta, G.: Graph drawing. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn, pp. 1163–1185. Chapman and Hall/CRC, London (2004). doi:10.1201/9781420035315.ch52
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Angelini, P., Bekos, M.A. (2017). Hierarchical Partial Planarity. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-68705-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68704-9
Online ISBN: 978-3-319-68705-6
eBook Packages: Computer ScienceComputer Science (R0)