Abstract
In this talk we digress about the strict interplay between the graph-theoretic problem of computing a Hamiltonian augmentation of a planar graph G and the graph drawing problem of embedding G onto a given set of points. We review different Hamiltonian augmentation techniques and their impact on different variants of the corresponding graph drawing problem. We also look at universal point sets, simultaneous graph embeddings, and radial graph drawings.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abellanas, M., Garcia-Lopez, J., Hernández-Peñver, G., Noy, M., Ramos, P.A.: Bipartite embeddings of trees in the plane. Discrete Applied Mathematics 93(2-3), 141–148 (1999)
Akiyama, J., Urrutia, J.: Simple alternating path problem. Discrete Mathematics 84, 101–103 (1990)
Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoretical Computer Science 408(2-3), 129–142 (2008)
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)
Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM Journal on Computing 14, 210–223 (1985)
Chiba, N., Nishizeki, T.: The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. Journal of Algorithms 10, 189–211 (1989)
Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)
Di Giacomo, E., Didimo, W., Liotta, G.: Radial drawings of graphs: Geometric constraints and trade-offs. Journal of Discrete Algorithms 6(1), 109–124 (2008)
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. Journal of Graph Algorithms and Applications 12(1), 29–49 (2008)
Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Computational Geometry 30, 1–23 (2005)
Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. Algorithmica (to appear)
Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. IJFCS, Special Issue on Graph Drawing 17(5), 1071–1094 (2006)
Di Giacomo, E., Liotta, G.: Simultaneous embedding of outerplanar graphs, paths, and cycles. International Journal of Computational Geometry and Applications 17(2), 139–160 (2007)
Enomoto, H., Miyauchi, M.S.: Embedding graphs into a three page book with O(m logn) crossings of edges over the spine. SIAM J. Discrete Math. 12(3), 337–341 (1999)
Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. Journal of Graph Algorithms and Applications 9(3), 347–364 (2005)
Everett, H., Lazard, S., Liotta, G., Wismath, S.K.: Universal sets of n points for 1-bend drawings of planar graphs with n vertices. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 345–351. Springer, Heidelberg (2008)
Giordano, F., Liotta, G., Mchedlidze, T., Symvonis, A.: Computing upward topological book embeddings of upward planar digraphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 172–183. Springer, Heidelberg (2007)
Giordano, F., Liotta, G., Whitesides, S.: Embeddability problems for upward planar digraphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 242–253. Springer, Heidelberg (2009)
Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Amer. Math. Monthly 98(2), 165–166 (1991)
Halton, J.H.: On the thickness of graphs of given degree. Information Sciences 54, 219–238 (1991)
Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Applied Mathematics 101, 167–175 (2000)
Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane - a survey. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete & Computational Geometry. Algorithms and Combinatories, vol. 25, pp. 551–570. Springer, Heidelberg (2003)
Kaneko, A., Kano, M., Suzuki, K.: Path coverings of two sets of points in the plane. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342. American Mathematical Society (2004)
Kaneko, A., Kano, M., Yoshimoto, K.: Alternating hamilton cycles with minimum number of crossing in the plane. International Journal of Computational Geometry & Application 10, 73–78 (2000)
Kaneko, A., Kano, M.: Straight-line embeddings of two rooted trees in the plane. Discrete & Computational Geometry 21(4), 603–613 (1999)
Kaneko, A., Kano, M., Tokunaga, S.: Straight-line embeddings of three rooted trees in the plane. In: Canadian Conference on Computational Geometry, CCCG 1998 (1998)
Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)
Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs. Inf. Process. Lett. 92(2), 95–98 (2004)
Mchedlidze, T., Symvonis, A.: Crossing-optimal acyclic hamiltonian path completion and its application to upward topological book embeddings. In: Das, S., Uehara, R. (eds.) WALCOM 2009. LNCS, vol. 5431, pp. 250–261. Springer, Heidelberg (2009)
Mchedlidze, T., Symvonis, A.: Crossing-optimal acyclic hp-completion for outerplanar t-digraphs. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 76–85. Springer, Heidelberg (2009)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph and Combinatorics 17, 717–728 (2001)
Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms (SODA 1990), pp. 138–148 (1990)
Sugiyama, K.: Graph Drawing and Applications. World Scientific, Singapore (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Di Giacomo, E., Liotta, G. (2010). The Hamiltonian Augmentation Problem and Its Applications to Graph Drawing. In: Rahman, M.S., Fujita, S. (eds) WALCOM: Algorithms and Computation. WALCOM 2010. Lecture Notes in Computer Science, vol 5942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11440-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-11440-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11439-7
Online ISBN: 978-3-642-11440-3
eBook Packages: Computer ScienceComputer Science (R0)