On Visibility Graphs — Upper Bounds and Classification of Special Types

We examine several types of visibility graphs: bar and semi-bar k-visibility graphs, rectangle k-visibility graphs, arc and circle k-visibility graphs, and compact visibility graphs. We improve the upper bound on the thickness of bar k-visibility graphs from 2k(9k − 1) to 6k, and prove that the upper bound must be at least k + 1. We also show that the upper bound on the thickness of semi-bar k-visibility graphs is between d23(k + 1)e and 2k. In addition, we give a method of finding the number of edges in a semi-bar k-visibility graph based on skyscraper puzzles. Analogous to bar and semi-bar k-visibility graphs, we establish bounds on the number of edges, chromatic number, and thickness of rectangle k-visibility graphs; as well as bounds on the number of edges and the chromatic number of arc and circle k-visibility graphs. Finally, we relate two conjectures on compact visibility graphs, prove that every n-partite graph in the form K1,a1,a2,...,an−1 is a compact visibility graph, and classify all (but one) graphs with at most six vertices as compact visibility graphs.