Visibility Graphs of Point Sets in the Plane

The visibility graph of a discrete point set X⊂ℝ² has vertex set X and an edge xy for every two points x,y∈X whenever there is no other point in X on the line segment between x and y. We show that for every graph G, there is a point set X∈ℝ², such that the subgraph of induced by X is isomorphic to G. As a consequence, we show that there are visibility graphs of arbitrary high chromatic number with clique number 6 settling a question by Kára, Pór and Wood. Supported by the DFG Research Center Matheon (FZT86).