Contact graph

In the mathematical area of graph theory, a contact graph or tangency graph is a graph whose vertices are represented by geometric objects (e.g. curves, line segments, or polygons), and whose edges correspond to two objects touching (but not crossing) according to some specified notion.[1] It is similar to the notion of an intersection graph but differs from it in restricting the ways that the underlying objects are allowed to intersect each other.

The circle packing theorem[2] states that every planar graph can be represented as a contact graph of circles. The contact graphs of unit circles are called penny graphs.[3] Representations as contact graphs of triangles,[4] rectangles,[5] squares,[6] line segments,[7] or circular arcs[8] have also been studied.

References
  1. Chaplick, Steven; G. Kobourov, Stephen; Ueckerdt, Torsten (2013-06-19). "Equilateral L-Contact Graphs". arXiv:1303.1279. online PDF
  2. Koebe, Paul (1936), "Kontaktprobleme der Konformen Abbildung", Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl., 88: 141–164
  3. Pisanski, Tomaž; Randić, Milan (2000), "Bridges between geometry and graph theory" (PDF), in Gorini, Catherine A. (ed.), Geometry at Work, MAA Notes, 53, Cambridge University Press, pp. 174–194, MR 1782654. See especially p. 176.
  4. de Fraysseix, Hubert; Ossona de Mendez, Patrice; Rosenstiehl, Pierre (1994), "On triangle contact graphs", Combinatorics, Probability and Computing, 3 (2): 233–246, doi:10.1017/S0963548300001139, MR 1288442
  5. Buchsbaum, Adam L.; Gansner, Emden R.; Procopiuc, Cecilia M.; Venkatasubramanian, Suresh (2008), "Rectangular layouts and contact graphs", ACM Transactions on Algorithms, 4 (1): Art. 8, 28, arXiv:cs/0611107, doi:10.1145/1328911.1328919, MR 2398588
  6. Klawitter, Jonathan; Nöllenburg, Martin; Ueckerdt, Torsten (2015), "Combinatorial properties of triangle-free rectangle arrangements and the squarability problem", Graph Drawing and Network Visualization: 23rd International Symposium, GD 2015, Los Angeles, CA, USA, September 24-26, 2015, Revised Selected Papers, Lecture Notes in Computer Science, 9411, Springer, pp. 231–244, arXiv:1509.00835, doi:10.1007/978-3-319-27261-0_20
  7. Hliněný, Petr (2001), "Contact graphs of line segments are NP-complete" (PDF), Discrete Mathematics, 235 (1–3): 95–106, doi:10.1016/S0012-365X(00)00263-6, MR 1829839
  8. Alam, Md. Jawaherul; Eppstein, David; Kaufmann, Michael; Kobourov, Stephen G.; Pupyrev, Sergey; Schulz, André; Ueckerdt, Torsten (2015), "Contact graphs of circular arcs", Algorithms and Data Structures: 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015, Proceedings, Lecture Notes in Computer Science, 9214, Springer, pp. 1–13, arXiv:1501.00318, doi:10.1007/978-3-319-21840-3_1.
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