Abstract
In this paper, we introduce Vertex-face contact representation (VFCR for short) for 2-connected plane multigraphs. We present a simple linear time algorithm for constructing a VFCR for 2-connected plane graphs. Our algorithm only uses an st-orientation for G and its corresponding st-orientation for the dual graph of G. We also show that one kind of vertex-vertex contact representation (VVCR) for 2-connected bipartite planar graphs introduced by Fraysseix et al. [2,3] can be easily obtained by applying our algorithm. In general, our algorithm produces a more compact representation than their algorithm.
Then we investigate st-orientations for 3-connected planar graphs. We prove that a 3-connected planar graph G with n vertices and f faces, has an st-orientation with the length of its longest directed path \(\leq \frac{2n}{3}+2\lceil\sqrt{n/3}\rceil+5\). This implies that such a graph G admits a VFCR in a grid with non-trivial size bound. This non-trivial size bound also applies to the vertex-vertex contact representation [2,3] for a large class of 2-connected bipartite planar graphs.
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References
Di Battista, G., Tamassia, R., Vismara, L.: Output-sensitive Reporting of Disjoint Paths. Algorithmica 23(4), 302–340 (1999)
Fraysseix, H.D., de Mendez, P.O., Pach, J.: Representation of planar graphs by segments. Intuitive Geometry 63, 109–117 (1991)
Fraysseix, H.D., de Mendez, P.O., Pach, J.: A Left-First Search Algorithm for Planar Graphs. Discrete & Computational Geometry 13, 459–468 (1995)
Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of Graphs Proc. of an International Symposium, Rome, July 1966, pp. 215–232 (1966)
Felsner, S.: Convex drawings of Planar Graphs and the Order Dimension of 3-Polytopes. Order 18, 19–37 (2001)
Miura, K., Azuma, M., Nishizeki, T.: Canonical decomposition, realizer, Schnyder Labelling and orderly spanning trees of plane graphs. International Journal of Foundations of Computer Science 16, 117–141 (2005)
Ossona de Mendez, P.: Orientations bipolaires, PhD thesis, Ecole des Hautes Etudes en Sciences Sociales, Paris (1994)
Papamanthou, C., Tollis, I.G.: Applications of parameterized st-orientations in graph drawings. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 355–367. Springer, Heidelberg (2006)
Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)
Tamassia, R., Tollis, I.G.: An unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)
He, X., Zhang, H.: Nearly Optimal Visibility Representations of Plane Graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 407–418. Springer, Heidelberg (2006)
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Sadasivam, S., Zhang, H. (2008). On Representation of Planar Graphs by Segments. In: Fleischer, R., Xu, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2008. Lecture Notes in Computer Science, vol 5034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68880-8_29
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DOI: https://doi.org/10.1007/978-3-540-68880-8_29
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