Equivalent nondegenerate L-shapes of double-loop networks

Double-loop networks have been widely studied as architecture for local area networks. The L-shape is an important tool for studying the distance properties of double-loop networks. Two L-shapes are equivalent if the numbers of nodes k steps away from the origin are the same for every k. Hwang and Xu first studied equivalent L-shapes through a geometric operation called 3-rectangle transformation. Fiol et al. proposed three equivalent transformations. Rodseth gave an algebraic operation, which was found by Huang et al. to correspond to 3-rectangle transformations. In this paper, we show that all equivalent nondegenerate L-shapes are determined by four basic geometric operations. We also discuss the algebraic operations corresponding to these geometric operations.