Approximate Proximity Drawings

Evans, Willliam and Gansner, Emden R. and Kaufmann, Michael and Liotta, Giuseppe and Meijer, Henk and Spillner, Andreas (2012) Approximate Proximity Drawings. In: Graph Drawing 19th International Symposium, GD 2011, September 21-23, 2011, Eindhoven, The Netherlands , pp. 166-178 (Official URL: http://dx.doi.org/10.1007/978-3-642-25878-7_17).

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Abstract

We introduce and study a generalization of the well-known region of influence proximity drawings, called (ε1,ε2)-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε1 ≥ 0 and ε2 ≥ 0, an (ε1,ε2)-proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u, v, their proximity region “shrunk” by the multiplicative factor 1:(1+ε1) does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u, v, their proximity region “blown-up” by the factor (1+ε2) contains some vertices of Γ other than u and v. We show that by using this generalization, we can significantly enlarge the family of the representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of ε1 and ε2 . We also study the extremal case of (0,ε2)-proximity drawings, which generalizes the well-known weak proximity drawing model.

Item Type:Conference Paper
Additional Information:10.1007/978-3-642-25878-7_17
Classifications:G Algorithms and Complexity > G.560 Geometry
ID Code:1251

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References

Angelini, P., Bruckdorfer, T., Chiesa, M., Frati, F., Kaufmann, M., Squarcella, C.: On the Area Requirements of Euclidean Minimum Spanning Trees. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 25–36. Springer, Heidelberg (2011)

Aronov, B., Dulieu, M., Hurtado, F.: Witness (Delaunay) graphs. Comput. Geom. 44(6-7), 329–344 (2011)

Aronov, B., Dulieu, M., Hurtado, F.: Witness Rectangle Graphs. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 73–85. Springer, Heidelberg (2011)

Bose, P., Lenhart, W., Liotta, G.: Characterizing proximity trees. Algorithmica 16(1), 83–110 (1996)

de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)

Di Battista, G., Lenhart, W., Liotta, G.: Proximity Drawability: a Survey. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 328–339. Springer, Heidelberg (1995)

Di Battista, G., Liotta, G., Whitesides, S.: The strength of weak proximity. J. Discrete Algorithms 4(3), 384–400 (2006)

Dillencourt, M.B.: Realizability of Delaunay triangulations. Inf. Process. Lett. 33(6), 283–287 (1990)

Dillencourt, M.B., Smith, W.D.: A Simple Method for Resolving Degeneracies in Delaunay Triangulations. In: Lingas, A., Carlsson, S., Karlsson, R. (eds.) ICALP 1993. LNCS, vol. 700, pp. 177–188. Springer, Heidelberg (1993)

Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H.: Drawing a Tree as a Minimum Spanning Tree Approximation. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 61–72. Springer, Heidelberg (2010)

Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)

Lenhart, W., Liotta, G.: Proximity Drawings of Outerplanar Graphs. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 286–302. Springer, Heidelberg (1997)

Li, X.: Applications of computational geometry in wireless networks. In: Cheng, X., Huang, X., Du, D.-Z. (eds.) Ad Hoc Wireless Networking, pp. 197–264. Kluwer Academic Publishers (2004)

Liotta, G.: Proximity drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. CRC Press (to appear)