Abstract
Since an image can easily be modeled by its adjacency graph, graph theory and algorithms on graphs are widely used in image processing. Of particular interest are the problems of estimating the number of the maximal cliques in a graph and designing algorithms for their computation, since these are found relevant to various applications in image processing and computer graphics. In the present paper we study the maximal clique problem on intersection graphs of convex polygons, which are also applicable to imaging sciences. We present results which refine or improve some of the results recently proposed in [18]. Thus, it was shown therein that an intersection graph of n convex polygons whose sides are parallel to k different directions has no more than n 2k maximal cliques. Here we prove that the number of maximal cliques does not exceed n k. Moreover, we show that this bound is tight for any fixed k. Algorithmic aspects are discussed as well.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ambühl, C., Wagner, U.: The clique problem in intersection graphs of ellipses and triangles. Theory Comput. Syst. 38, 279–292 (2005)
Ballard, D.H., Brown, M.: Computer Vision. Prentice-Hall, Englewood Cliffs (1982)
Brimkov, V.E., Kafer, S., Szczepankiewicz, M., Terhaar, J.: Maximal cliques in intersection graphs of quasi-homothetic trapezoids. In: Proc. MCURCSM 2013, Ohio, 10 p. (2013)
Cabello, S., Cardinal, J., Langerman, S.: The clique problem in ray intersection graphs. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 241–252. Springer, Heidelberg (2012)
Čulík, K.: Applications of graph theory to mathematical logic and linguistics. In: Proc. Sympos. “Theory of Graphs and its Applications” (Smolenice, 1963), pp. 13–20. Publ. House Czechoslovak Acad. Sci., Prague (1964)
Erdős, P, Goodman, A.W., Pósa, L.: The representation of a graph by set intersections. Canad. J. Math. 18, 106–112 (1966)
Evako, A.V.: Topological properties of the intersection graph of covers of n-dimensional surfaces. Discrete Mathematics 147(1-3), 107–120 (1995)
Felsner, S., Müller, R., Wernisch, L.: Trapezoid graphs and generalizations, geometry and algorithms. Discrete Applied Mathematics 74, 13–32 (1993)
Fish, A., Stapleton, G.: Formal issues in languages based on closed curves. In: Proc. Distributed Multimedia Systems, pp. 161–167 (2006)
Gardiner, E.J., Artymiuk, P.J., Willett, P.: Clique-detection algorithms for matching three-dimensional molecular structures. J. Molecular Graph Modelling 15(4), 245–253 (1997)
Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs. Acad. Press (1980)
Heinzle, F., Ander, K.H., Sester, M.: Graph based approaches for recognition of patterns and implicit information in road networks. In: Proc. 22nd International Cartographic Conference, A Coruna (2005)
Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. Journal of Algorithms 4, 300–323 (1983)
Ion, A., Carreira, J., Sminchisescu, C.: Image segmentation by figure-ground composition into maximal cliques. In: Proc. 13th International Conference on Computer Vision, Barcelona, pp. 2110–2117 (2011)
Jacobson, M.S., Morris, F.R., Scheinermann, E.R.: General results on tolerance intersection graphs. J. Graph Theory 15, 573–577 (1991)
Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Information Processing Letters 27, 119–123 (1988)
Klette, R., Rosenfeld, A.: Digital Geometry. Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)
Junosza-Szaniawski, K., Kratochvíl, J., Pergel, M., Rzążewski, P.: Beyond homothetic polygons: Recognition and maximum clique. In: Chao, K.-M., Hsu, T.-s., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 619–628. Springer, Heidelberg (2012)
Kaufmann, M., Kratochvíl, J., Lehmann, K., Subramanian, A.: Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition. In: Proc. SODA 2006, pp. 832–841 (2006)
Kratochvíl, J., Kuběna, A.: On intersection representations of co-planar graphs. Discrete Mathematics 178, 251–255 (1998)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combinatorial Theory Ser. B 62, 289–315 (1994)
Kratochvíl, J., Nešetřil, J.: Independent set and clique problems in intersection-defined classes of graphs. Comm. Math. Uni. Car. 31, 85–93 (1990)
Kratochvíl, J., Pergel, M.: Intersection graphs of homothetic polygons. Electronic Notes in Discr. Math. 31, 277–280 (2008)
McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications, vol. 2. SIAM, Philadelphia (1999)
Szpilrajn-Marczewski, E.: Sur deux propriétés des classes d’ensembles. Fund. Math. 33, 303–307 (1945)
Nakamura, H., Masatake, H., Mamoru, H.: Robust computation of intersection graph between two solids. Graphical Models 16(3), C79–C88 (1997)
Nandy, S.C., Bhattacharya, B.B.: A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids. Computers Math. Applic. 29(8), 45–61 (1995)
Paget, R., Longsta, D.: Extracting the cliques from a neighbourhood system. IEE Proc. Vision Image and Signal Processing 144(3), 168–170 (1997)
Simonetto, P., Auber, D.: An heuristic for the construction of intersection graphs. In: Proc. 13th International Conference on Information Visualisation, pp. 673–678 (2009)
Simonetto, P., Auber, D.: Visualise undrawable Euler diagrams. In: Proc. 12th IEEE International Conference on Information Visualisation, pp. 594–599 (2008)
Tian, J., Tinghua, A., Xiaobin, J.: Graph based recognition of grid pattern in street networks. In: The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Advances in Spatial Data Handling and GIS. Lecture Notes in Geoinformation and Cartography, Part II, vol. 38, pp. 129–143 (2012)
Vairinhos, V.M., Lobo, V., Galindo, M.P.: Intersection graph-based representation of contingency tables, http://www.isegi.unl.pt/docentes/vlobo/Publicacoes/3_17_lobo08_DAIG_conting_tables.pdf
Müller, T., van Leeuven, E.J., van Leeuven, J.: Integer representations of convex polygons intersection graphs. In: Symposium on Computational Geometry, pp. 300–307 (2011)
van Leeuwen, E.J., van Leeuwen, J.: Convex polygon intersection graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 377–388. Springer, Heidelberg (2011)
Verroust, A., Viaud, M.-L.: Ensuring the drawability of extended euler diagrams for up to 8 sets. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 128–141. Springer, Heidelberg (2004)
Wang, X., Bai, X., Yang, X., Wenyu, L., Latecki, L.J.: Maximal cliques that satisfy hard constraints with application to deformable object model learning. In: Advances in Neural Information Processing Systems, vol. 24, pp. 864–872 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Brimkov, V.E., Kafer, S., Szczepankiewicz, M., Terhaar, J. (2014). On Intersection Graphs of Convex Polygons. In: Barneva, R.P., Brimkov, V.E., Šlapal, J. (eds) Combinatorial Image Analysis. IWCIA 2014. Lecture Notes in Computer Science, vol 8466. Springer, Cham. https://doi.org/10.1007/978-3-319-07148-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-07148-0_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07147-3
Online ISBN: 978-3-319-07148-0
eBook Packages: Computer ScienceComputer Science (R0)