Does a graph of a circle represent a graph of a function? - Answers

Although closely related problems in discrete geometry had been studied earlier, e.g. by Scott (1970) and Jamison (1984), the problem of determining the slope number of a graph was introduced byWade & Chu (1994), who showed that the slope number of an n-vertex complete graph Knis exactly n. A drawing with this slope number may be formed by placing the vertices of the graph on a regular polygon. As Keszegh, Pach & Pálvölgyi (2011) showed, every planar graph has a planar straight-line drawing in which the number of distinct slopes is a function of the degree of the graph. Their proof follows a construction of Malitz & Papakostas (1994) for bounding the angular resolution of planar graphs as a function of degree, by completing the graph to a maximal planar graph without increasing its degree by more than a constant factor, and applying the circle packing theorem to represent this augmented graph as a collection of tangent circles. If the degree of the initial graph is bounded, the ratio between the radii of adjacent circles in the packing will also be bounded, which in turn implies that using a quadtree to place each graph vertex on a point within its circle will produce slopes that are ratios of small integers. The number of distinct slopes produced by this construction is exponential in the degree of the graph.