Computing Radial Drawings on the Minimum Number of Circles

A radial drawing is a representation of a graph in which the vertices lie on concentric circles of finite radius. In this paper we study the problem of computing radial drawings of planar graphs by using the minimum number of concentric circles. We assume that the edges are drawn as straight-line segments and that co-circular vertices can be adjacent. It is proven that the problem can be solved in polynomial time. The solution is based on a characterization of those graphs that admit a crossing-free straight-line radial drawing on k circles. For the graphs in this family, a linear time algorithm that computes a radial drawing on k circles is also presented. Article Type Communicated by Submitted Revised Regular Paper E. R. Gansner and J. Pach January 2005 July 2005 Research partially supported by MIUR under Project ALGO-NEXT (Algorithms for the Next Generation Internet and Web: Methodologies, Design and Experiments), and by NSERC. An extended abstract of this paper was presented at 12 International Symposium on Graph Drawing, GD 2004. E. Di Giacomo et al., Radial Drawings, JGAA, 9(3) 365–389 (2005) 366