Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
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The visibility graph $\mathcal {V}(X)$ 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 $\mathcal {V}(X\cup \mathbb {Z}^{2})$ 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.