Cover Contact Graphs

Atienza, Nieves and De Castro, Natalia and Cortés, Carmen and Garrido, M. Ángeles M. Ángeles and Grima, Clara I. and Hernández, Gregorio and Márquez, Alberto and Moreno, Auxiliadora and Nöllenburg, Martin and Portillo, José Ramon and Reyes, Pedro and Valenzuela, Jesús and Villar, Maria Trinidad and Wolff, Alexander (2008) Cover Contact Graphs. In: Graph Drawing 15th International Symposium, GD 2007, September 24-26, 2007, Sydney, Australia , pp. 171-182 (Official URL: http://dx.doi.org/10.1007/978-3-540-77537-9_18).

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Abstract

We study problems that arise in the context of covering certain geometric objects (so-called $seeds$, e.g., points or disks) by a set of other geometric objects (a so-called $cover$, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a $cover contact graph$ (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).

Item Type:Conference Paper
Additional Information:10.1007/978-3-540-77537-9_18
Classifications:Z Theory > Z.250 Geometry
ID Code:836

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References

M. Abellanas, S. Bereg, F. Hurtado, A. G. Olaverri, D. Rappaport, and J. Tejel. Moving coins. Comput. Geom. Theory Appl., 34(1):35-48, 2006.

M. Abellanas, N. de Castro, G. Hernández, A. Márquez, and C. Moreno-Jiménez. Gear system graphs. Manuscript, 2006.

N. Atienza, N. de Castro, C. Cortés, M. Á. Garrido, C. I. Grima, G. Hernández, A. Márquez, A. Moreno, M. Nöllenburg, J. R. Portillo, P. Reyes, J. Valenzuela, M. T. Villar, and A. Wolff. Cover contact graphs. Technical Report 2007-18, Fakultät für Informatik, Universität Karlsruhe, Sept. 2007. Available at http://www.ubka.uni-karlsruhe.de/indexer-vvv/ira/2007/18.

C. R. Collins and K. Stephenson. A circle packing algorithm. Comput. Geom. Theory Appl., 25(3):233-256, 2003.

H. de Fraysseix and P. O. de Mendez. Representations by contact and intersection of segments. Algorithmica, 47(4):453-463, 2007.

S. Fortune. A sweepline algorithm for voronoi diagrams. In Proc. 2nd Annu. Sympos. Comput. Geom. (SoCG'86), pages 313-322, 1986.

O. Giménez and M. Noy. The number of planar graphs and properties of random planar graphs. In C. Martínez, editor, Proc. Internat. Conf. Anal. Algorithms (ICAA'05), volume AD of DMTCS Proceedings, pages 147-156, 2005.

M. Kaufmann and R. Wiese. Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl., 6(1):115-129, 2002.

P. Koebe. Kontaktprobleme der konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Klasse, 88:141-164, 1936.

J. S. B. Mitchell. L1 shortest paths among polygonal obstacles in the plane. Algorithmica, 8:55-88, 1992.

J. Pach and P. K. Agarwal. Combinatorial Geometry. John Wiley and Sons, New York, 1995. contains a proof of Koebe's theorem.

H. Sachs. Coin graphs, polyhedra, and conformal mapping. Discrete Math., 134(1-3):133-138, 1994.

W. P. Thurston. The Geometry and Topology of 3-Manifolds. Princeton University Notes, 1980.

G. T. Toussaint. A graph-theoretical primal sketch. In G. T. Toussaint, editor, Computational Morphology: A Computational Geometric Approach to the Analysis of Form, pages 229-260. North-Holland, 1988.

E. Welzl. Smallest enclosing disks (balls and ellipsoids). In H. Maurer, editor, New Results and New Trends in Computer Science, volume 555 of Lecture Notes Comput. Sci., pages 359-370. Springer-Verlag, 1991.