Abstract
In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x,y in P are adjacent whenever the open rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers.
We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs.
Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.
Research of B.A. has been partially supported by NSA MSP Grants H98230-06-1-0016 and H98230-10-1-0210. Research of B.A. and M.D. has also been supported by a grant from the U.S.-Israel Binational Science Foundation and by NSF Grant CCF-08-30691. Research by F.H. has been partially supported by projects MEC MTM2006-01267 and MTM2009-07242, and Gen. Catalunya DGR 2005SGR00692 and 2009SGR1040.
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.
References
Aronov, B., Dulieu, M., Hurtado, F.: Witness (Delaunay) graphs. Computational Geometry Theory and Applications 44(6-7), 329–344 (2011)
Aronov, B., Dulieu, M., Hurtado, F.: Witness Gabriel graphs. Computational Geometry Theory and Applications (to appear)
Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J., Urrutia, G. (eds.) Handbook of Computational Geometry, ch.5, pp. 201–290. Elsevier Science Publishing, Amsterdam (2000)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1998)
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)
de Berg, M., Carlsson, S., Overmars, M.: A general approach to dominance in the plane. J. Algorithms 13(2), 274–296 (1992)
Chartrand, G., Lesniak, L.: Graphs and Digraphs, 4th edn. Chapman & Hall, Boca Raton (2004)
Collette, S.: Regions, Distances and Graphs., PhD thesis. Université Libre de Bruxelles (2006)
Czyzowicz, J., Kranakis, E., Urrutia, J.: Dissections, cuts, and triangulations. In: Proc. 11th Canadian Conf. Comput. Geometry, pp. 154–157 (1999)
Dillencourt, M.B.: Toughness and Delaunay triangulations. Discrete Comput. Geometry 5(6), 575–601 (1990)
Dillencourt, M.B.: Realizability of Delaunay triangulations. Inf. Proc. Letters 33(6), 283–287 (1990)
Dillencourt, M.B., Smith, W.D.: Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Math. 161(1-3), 63–77 (1996)
Dulieu, M.: Witness proximity graphs. PhD thesis. Polytechnic Institute of NYU, Brooklyn, New York (2012)
Dushnik, B., Miller, E.W.: Partially ordered sets. American J. Math. 63, 600–619 (1941)
Erdős, P., Fajtlowicz, S., Hoffman, A.J.: Maximum degree in graphs of diameter 2. Networks 10(1), 87–90 (2006)
Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., ch.23, pp. 513–528. CRC Press, Boca Raton (2004)
Habib, M., McConnell, R., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science 234(1-2), 59–84 (2000)
Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)
Katchalski, M., Meir, A.: On empty triangles determined by points in the plane. Acta Math. Hungar. 51, 23–328 (1988)
Liotta, G.: Proximity Drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. Chapman & Hall/CRC Press, (in preparation)
Liotta, G., Lubiw, A., Meijer, H., Whitesides, S.: The rectangle of influence drawability problem. Computational Geometry Theory and Applications 10(1), 1–22 (1998)
McConnell, R.: Personal communication (2011)
McKay, B.D., Miller, M., Širáň, J.: A note on large graphs of diameter two and given maximum degree. Combin. Theory Ser. B 74, 110–118 (1998)
Miller, M., Širáň, J.: Moore graphs and beyond: A survey of the degree/diameter problem. Electron. J. Combin. DS14, 61 (2005)
Sakai, T., Urrutia, J.: Covering the convex quadrilaterals of point sets. Graphs and Combinatorics 23(1), 343–357 (2007)
Toussaint, G.T. (ed.): Computational Morphology. North-Holland, Amsterdam (1988)
Weisstein, E.W.: Graph Join. From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/GraphJoin.html
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aronov, B., Dulieu, M., Hurtado, F. (2011). Witness Rectangle Graphs. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-22300-6_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22299-3
Online ISBN: 978-3-642-22300-6
eBook Packages: Computer ScienceComputer Science (R0)