Abstract
We introduce and study the problem Mutual Planar Duality, which asks for planar graphs G 1 and G 2 whether G 1 can be embedded such that its dual is isomorphic to G 2. We show NP-completeness for general graphs and give a linear-time algorithm for biconnected graphs.
We consider the common dual relation ~, where G 1 ~G 2 if and only they admit embeddings that result in the same dual graph. We show that ~ is an equivalence relation on the set of biconnected graphs and devise a succinct, SPQR-tree-like representation of its equivalence classes. To solve Mutual Planar Duality for biconnected graphs, we show how to do isomorphism testing for two such representations in linear time.
A special case of Mutual Planar Duality is testing whether a graph is self-dual. Our algorithm can handle the case of biconnected graphs in linear time and our NP-hardness proof extends to self-duality and also to map self-duality testing (which additionally requires to preserve the embedding).
Partially supported by ESF project 10-EuroGIGA-OP-003 GraDR “Graph Drawings and Representations”. This work began during a visit of Angelini at Karlsruhe Institute of Technology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Angelini, P., Di Battista, G., Patrignani, M.: Finding a minimum-depth embedding of a planar graph in O(n 4) time. Algorithmica 60, 890–937 (2011)
Angelini, P., Bläsius, T., Rutter, I.: Testing mutual duality of planar graphs. CoRR abs/1303.1640 (2013)
Archdeacon, D., Richter, R.B.: The construction and classification of self-dual spherical polyhedra. Journal of Combinatorial Theory, Series B 54(1), 37–63 (1992)
Bienstock, D., Monma, C.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5, 93–109 (1990)
Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996)
Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
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)
Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs (preliminary report). In: Proceedings of the 6th Annual ACM Symposium on Theory of Computing (STOC 1974), pp. 172–184. ACM (1974)
Raghavendra Rao, B., Jayalal Sarma, M.: On the complexity of matroid isomorphism problem. Theory of Computing Systems 49(2), 246–272 (2011)
Servatius, B., Christopher, P.R.: Construction of self-dual graphs. Am. Math. Monthly 99(2), 153–158 (1992)
Servatius, B., Servatius, H.: Self-dual graphs. Discrete Math. 149(1-3), 223–232 (1996)
Tutte, W.T.: Connectivity in matroids. Canad. J. Math. 18, 1301–1324 (1966)
Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54(1), 150–168 (1932)
Whitney, H.: 2-isomorphic graphs. Amer. J. Math. 55, 245–254 (1933)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Angelini, P., Bläsius, T., Rutter, I. (2013). Testing Mutual Duality of Planar Graphs. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-45030-3_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-45029-7
Online ISBN: 978-3-642-45030-3
eBook Packages: Computer ScienceComputer Science (R0)