Orthogonal 3D Shapes of Theta Graphs

The recent interest in three dimensional graph drawing has been motivating studies on how to extend two dimensional techniques to higher dimensions. A common approach for computing a 2D orthogonal drawing ofa graph separates the task of defining the shape ofthe drawing from the task of computing its coordinates. First results towards finding a three-dimensional counterpart of this approach are presented in [8,9], where characterizations of orthogonal representations of paths and cycles are studied. In this note we show that the known characterization for cycles does not immediately extend to even seemingly simple graphs such as theta graphs. A sufficient condition for recognizing three-dimensional orthogonal representations oftheta graphs is also presented.