On colouring point visibility graphs

Abstract In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4.

[1]  Ajit A. Diwan,et al.  Partitions of planar point sets into polygons , 2016, CCCG.

[2]  Jean Cardinal,et al.  Recognition and Complexity of Point Visibility Graphs , 2015, Discret. Comput. Geom..

[3]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[4]  Subir Kumar Ghosh,et al.  Unsolved problems in visibility graphs of points, segments, and polygons , 2010, ACM Comput. Surv..

[5]  Tomás Lozano-Pérez,et al.  An algorithm for planning collision-free paths among polyhedral obstacles , 1979, CACM.

[6]  Jan Mycielski.  Sur le coloriage des graphs , 1955 .

[7]  Raimund Seidel,et al.  Constructing Arrangements of Lines and Hyperplanes with Applications , 1986, SIAM J. Comput..

[8]  Florian Pfender.  Visibility Graphs of Point Sets in the Plane , 2008, Discret. Comput. Geom..

[9]  Subir Kumar Ghosh,et al.  Visibility Algorithms in the Plane , 2007 .

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  David R. Wood,et al.  On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane , 2005, Discret. Comput. Geom..

[12]  Bernard Chazelle,et al.  The power of geometric duality , 1985, BIT Comput. Sci. Sect..

[13]  Jan Kyncl,et al.  On planar point sets with the pentagon property , 2013, SoCG '13.

[14]  Subir Kumar Ghosh,et al.  Some results on point visibility graphs , 2015, Theor. Comput. Sci..

[15]  Bodhayan Roy,et al.  Point Visibility Graph Recognition is NP-Hard , 2014, Int. J. Comput. Geom. Appl..

[16]  David R. Wood,et al.  On the Connectivity of Visibility Graphs , 2011, Discret. Comput. Geom..