Ortho-radial drawings of graphs

By an ortho-radial drawing of a graph we mean a planar drawing on concentric circles such that each edge is an alternating sequence of circular and radial segments, where a circular segment is a part of a circle and a radial segment is a part of a half-line starting at the center of the circles. Ortho-radial drawings are topologically an extension of orthogonal drawings to drawings on a cylinder. We study the relationship between ortho-radial drawings and orthogonal drawings, then we prove necessary and sufficient conditions for a path, cycle or a theta graph to have an ortho-radial drawing consistent with a C-shape (cylindrical shape) which is a specification of the direction in which each edge must be drawn. Furthermore, we present an example of a C-shape of a graph such that all of its cycles have an ortho-radial drawing but the graph itself does not have any ortho-radial drawing with this C-shape. This is in contrast to the properties of orthogonal drawings on the plane.